--- slug: ulam-spiral passed: 2026-10-14 by: the assay line (research/assay-line/cited-1014b/), Codex fixer directed by claude-assaying-boyle-1014; source record by Codex, checked by Codex and a Claude agent checked: 212 confirmed: 137 wrong: 72 unverifiable: 1 observed: 0 stale: 2 --- This pass lands the directing instance’s 75 triage items for ulam-spiral read at fa98ee8ee1, after the fixer reread the supporting evidence and reran the affected arithmetic. It preserves the checkers’ remaining confirmed claims at their supported scope, with duplicate confirmations merged by source-record entry and current wording used where the same passage changed. The source record and two checker records are in research/assay-line/cited-1014b/ulam-spiral/. This is a directed fix pass, not a new blind check; the “went home” disagreement is retained as UNVERIFIABLE. All MINOR lines count toward wrong and receive a finding, but no dated page line. The two JSON-LD STALE items use the explicit metadata exception. Repair by Codex, 2026-10-14: R1 records the fix-checker's shared research copies of the global-diagonal claim as WRONG and open pending a board note. The authored layer corrections remain fixed; the shared inputs are outside this worker's edit scope. ## Claims - WRONG | X1: At a scientific meeting in 1963, sitting through what he later called “the presentation of a long and very boring paper,” Stanisław Ulam began to doodle. | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf | Gardner p.122 quotes only “long and very boring paper”. The added words are outside that quoted span in the source. Restored the span, with connective prose and comma outside the quotation. - WRONG | X2: 2, 3, 5, 7, 11, 13, no formula, no rhythm anyone has found in two thousand years of looking. | https://arxiv.org/abs/2004.01285 | Elsholtz’s abstract gives unconditional prime-producing formulas. Removed the blanket denial; the page now describes irregular prime gaps. - WRONG | X3: they grow by a fixed, widening stride, the signature of a quadratic. | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf; research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | Gardner p.122 qualifies lines beyond their initial spiral segment. Rebuilt the outward offset ray (1,-1)+k(1,1): 9,11,29,55,89,131, with second differences 16,8,8,8. Restricted the centre formulas to outward rays and qualified offset and axial rays as eventually quadratic. - WRONG | X4: Every diagonal of the spiral, central or offset, is one of these parabolic strings. | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf; research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | Gardner p.122 qualifies lines beyond their initial spiral segment. Rebuilt the outward offset ray (1,-1)+k(1,1): 9,11,29,55,89,131, with second differences 16,8,8,8. Restricted the centre formulas to outward rays and qualified offset and axial rays as eventually quadratic. - WRONG | X5: A horizontal or vertical line of the spiral is also a quadratic, but the diagonals are the ones whose values are always odd | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf; research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | Gardner’s checkerboard account and the actual walk give an even diagonal 2,12,30,56,90,132,182. Diagonals retain one parity; rows and columns alternate. Replaced the all-odd claim. - MINOR | X6: The primes have nowhere to live but the odd diagonals. | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf | Gardner p.122 explicitly excepts 2. Added the exception to the summary and markdown dek; no new dated correction for this MINOR item. - WRONG | X7: A random odd number near 1000 is prime about one time in seven | https://www.math.ucla.edu/~tao/preprints/Slides/austms.pdf; research/ulam-spiral/verify.mjs | Tao p.11 gives the odd-integer baseline 2/log N. Recomputed 2/log(1000)=0.289529654602 and counted 30 primes among the 100 odd integers 901..1099. Changed one in seven to about one in three and a half. - MINOR | X8: The answer, found by Georg Rabinowitsch in 1913, reaches into a different country of mathematics entirely. | https://www.impan.pl/shop/en/publication/transaction/download/product/82930 | Granville and Mollin p.139 names the 1912 ICM; reference 22 dates the proceedings 1913. Changed “found ... in 1913” to presented in 1912 and published in 1913; no new dated line. - WRONG | X9: The polynomial n²+n+c is prime for all of n = 0 … c−2 if and only if the field ℚ(√(1−4c)) has class number 1. | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | Granville and Mollin p.139 requires squarefree 4c-1 and field class number one. At c=7, the field is Q(sqrt(-3)) with class number one but f(1)=9. Added squarefreeness and integer c>=2, including the full-run metaphor and six-constant characterization. - WRONG | X10: is exactly, with no slack, the invisible miracle of factorization staying unique in a hidden number system. | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | Granville and Mollin p.139 requires squarefree 4c-1 and field class number one. At c=7, the field is Q(sqrt(-3)) with class number one but f(1)=9. Added squarefreeness and integer c>=2, including the full-run metaphor and six-constant characterization. - WRONG | X11: The lab computes that class number live, by counting reduced quadratic forms, and reports the verdict in both languages at once. | research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html; https://www.impan.pl/shop/en/publication/transaction/download/product/82930 | The original form counter gives order h(-75)=2 and h(-99)=2, while the corresponding fields have fundamental discriminants -3 and -11 and class number one. The lab now reduces to the field discriminant before counting forms and preserves its separate squarefree test. - CONFIRMED | X12: So the question “is there a polynomial luckier than Euler’s?” becomes “is there an imaginary quadratic field of class number one with a more negative discriminant than −163?” | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; public/strata/ulam-spiral/index.html | REJECTED with evidence: the preceding paragraph defines lucky as a full run n=0..c-2. Granville and Mollin p.139 classifies that property, whose largest c>=2 is 41. The local heading/question is valid in that sense; an arbitrary-streak counterexample addresses another question. Left this wording intact. - MINOR | X13: The full list of d for which ℚ(√−d) has class number one is one of the famous finite lists in mathematics | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; https://math.nyu.edu/~tschinke/books/gauss-dirichlet/stark.pdf | Multiplying d by a square leaves Q(sqrt(-d)) unchanged. The finite list uses positive squarefree d. Added that domain without changing the nine entries; no new dated line. - WRONG | X14: it is provably the best one of its entire kind that can ever exist. | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; research/ulam-spiral/verify.mjs | Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - WRONG | X15: The forty-prime run is a world record that the structure of mathematics forbids anyone from breaking. | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; research/ulam-spiral/verify.mjs | Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - WRONG | X16: Rabinowitsch settles the unbroken opening run | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; research/ulam-spiral/verify.mjs | Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - MINOR | X17: for the density of primes taken by any quadratic. | https://www.ams.org/journals/mcom/2003-72-241/S0025-5718-02-01418-7/S0025-5718-02-01418-7.pdf | Jacobson and Williams p.499 states positive leading coefficient, primitive coefficients, nonsquare discriminant and no parity obstruction. The prose now specifies irreducibility and no fixed prime divisor, with positive leading coefficient; no new dated line. - MINOR | X18: Every number on this page is checked; this is the one we mark as unknown, because that is what it is. | research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | The verifier performs finite arithmetic checks; the long near-integer decimal is a literal in the browser script. Narrowed prose to the listed checks and distinguished live calculations from supplied figures; no new dated line. The expanded arithmetic verifier passes 23/23. - WRONG | X19: why the opening run is exactly forty, why no parabola will ever beat it | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; research/ulam-spiral/verify.mjs | Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - MINOR | X20: Every claim above is recomputed offline in research/ulam-spiral/verify.mjs before the page asserts it, and the browser re-derives the live figures from scratch. | research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | The verifier performs finite arithmetic checks; the long near-integer decimal is a literal in the browser script. Narrowed prose to the listed checks and distinguished live calculations from supplied figures; no new dated line. The expanded arithmetic verifier passes 23/23. - MINOR | X21: n²+n+41 is prime for n=0…39 (40 in a row), composite at n=40 = 41²; prime for 4,149 of the first 10,001 values (≈41%). | research/ulam-spiral/verify.mjs | The computation is f(40)=1681=41²; 40 is the input. Corrected the two compressed input/output equalities, without a new dated line. - MINOR | X22: The honesty bar: every number checked offline and re-derived live; the one open question named as open. | research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | The verifier performs finite arithmetic checks; the long near-integer decimal is a literal in the browser script. Narrowed prose to the listed checks and distinguished live calculations from supplied figures; no new dated line. The expanded arithmetic verifier passes 23/23. - WRONG | X23: every diagonal of the spiral is a quadratic 4n²+bn+c, and the all-odd ones are the only place a prime can live. | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf; research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | Gardner p.122 qualifies lines beyond their initial spiral segment. Rebuilt the outward offset ray (1,-1)+k(1,1): 9,11,29,55,89,131, with second differences 16,8,8,8. Restricted the centre formulas to outward rays and qualified offset and axial rays as eventually quadratic. - WRONG | X24: The richest prime-making parabola anyone has found is Euler's n²+n+41 | https://www.ams.org/journals/mcom/2003-72-241/S0025-5718-02-01418-7/S0025-5718-02-01418-7.pdf | Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run. - MINOR | X25: which yields 40 primes in a row (n=0…39) and breaks exactly at n=40 = 41² | research/ulam-spiral/verify.mjs | The computation is f(40)=1681=41²; 40 is the input. Corrected the two compressed input/output equalities, without a new dated line. - WRONG | X26: Rabinowitsch's theorem (1913) says, with no slack, that the full run happens precisely when the imaginary quadratic field ℚ(√(1−4c)) has unique factorization, i.e. class number 1. | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | Granville and Mollin p.139 requires squarefree 4c-1 and field class number one. At c=7, the field is Q(sqrt(-3)) with class number one but f(1)=9. Added squarefreeness and integer c>=2, including the full-run metaphor and six-constant characterization. - WRONG | X27: So Euler's lucky numbers 2,3,5,11,17,41 are exactly the primes whose discriminants −7,−11,−19,−43,−67,−163 are the nine Heegner numbers | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; https://math.nyu.edu/~tschinke/books/gauss-dirichlet/stark.pdf; public/strata/ulam-spiral/index.html | The six constants map to discriminants -7,-11,-19,-43,-67,-163. The remaining class-number-one fields have discriminants -3,-4,-8. Corrected the identification: minus six of nine positive Heegner numbers, not the same census. - WRONG | X28: 163 is the last, so Euler's 40-prime run is a record the structure of mathematics forbids anyone to break | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; research/ulam-spiral/verify.mjs | Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - MINOR | X29: the same 163 is why e^{π√163} is an integer to twelve decimals | public/strata/ulam-spiral/index.html; research/assay-line/cited-1014b/ulam-spiral/N2-claude.md | Independent decimal arithmetic at 80-digit working precision gives a gap 7.4992740280181431112e-13. That is within 10^-12, but does not round to an integer at twelve decimal places. Changed the dek to “within 10^-12 of an integer”; no new dated line. - WRONG | X30: An immersive instrument: the live spiral with primes lit and each diagonal's formula fitted; a polynomial lab that runs n²+n+c, lights its primes, and computes the class number live to deliver Rabinowitsch's verdict in both languages; the Heegner census table | research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html; https://www.impan.pl/shop/en/publication/transaction/download/product/82930 | The original form counter gives order h(-75)=2 and h(-99)=2, while the corresponding fields have fundamental discriminants -3 and -11 and class number one. The lab now reduces to the field discriminant before counting forms and preserves its separate squarefree test. - MINOR | X31: Hardy–Littlewood Conjecture F (1923) predicts the diagonals' brightness to 6.64× and matches the count | https://www.ams.org/journals/mcom/2003-72-241/S0025-5718-02-01418-7/S0025-5718-02-01418-7.pdf; research/ulam-spiral/verify.mjs | The factor is for Euler’s polynomial compared with random integers of the same size. The verifier measures 6.6412 through n=1,000,000. Restricted the dek to that polynomial and baseline; no new dated line. - WRONG | X32: Each diagonal is a quadratic, and the richest anyone has found is Euler's n²+n+41, which yields 40 primes in a row. | https://www.ams.org/journals/mcom/2003-72-241/S0025-5718-02-01418-7/S0025-5718-02-01418-7.pdf | Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run. - WRONG | X33: A live spiral and a polynomial lab show, through Rabinowitsch's theorem and the Heegner numbers, why that record cannot be beaten. | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; research/ulam-spiral/verify.mjs | Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - WRONG | X34: it lays Euler's lucky numbers beside the nine Heegner numbers and shows them to be the same census | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; https://math.nyu.edu/~tschinke/books/gauss-dirichlet/stark.pdf; public/strata/ulam-spiral/index.html | The six constants map to discriminants -7,-11,-19,-43,-67,-163. The remaining class-number-one fields have discriminants -3,-4,-8. Corrected the identification: minus six of nine positive Heegner numbers, not the same census. - MINOR | X35: Every figure is recomputed offline in `research/ulam-spiral/verify.mjs` (17/17) before the page asserts it, and re-derived from scratch in your browser. | research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | The verifier performs finite arithmetic checks; the long near-integer decimal is a literal in the browser script. Narrowed prose to the listed checks and distinguished live calculations from supplied figures; no new dated line. The expanded arithmetic verifier passes 23/23. - WRONG | X36: the richest quadratic, Euler's n²+n+41, owes its 40 primes to the last of the nine Heegner numbers. | https://www.ams.org/journals/mcom/2003-72-241/S0025-5718-02-01418-7/S0025-5718-02-01418-7.pdf | Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run. - WRONG | X37: the proof that every diagonal is a quadratic 4n²+bn+c | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf; research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | Gardner p.122 qualifies lines beyond their initial spiral segment. Rebuilt the outward offset ray (1,-1)+k(1,1): 9,11,29,55,89,131, with second differences 16,8,8,8. Restricted the centre formulas to outward rays and qualified offset and axial rays as eventually quadratic. - MINOR | X38: Primes lie thick on the diagonals, sparse on the horizontals and verticals. | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf | Gardner p.122 also describes vertical and horizontal clusters, with alternating odd cells. Qualified both the initial and reset readouts as especially visible odd-diagonal streaks, with clusters in other orientations; no new dated line. - MINOR | X39: Euler’s lucky numbers are the Heegner discriminants, verified live | public/strata/ulam-spiral/index.html; https://www.impan.pl/shop/en/publication/transaction/download/product/82930 | The table distinguishes c from 1-4c. Its caption now describes a correspondence between six constants and six field discriminants; no new dated line. - WRONG | X40: The most prime-rich parabola anyone has found | https://www.ams.org/journals/mcom/2003-72-241/S0025-5718-02-01418-7/S0025-5718-02-01418-7.pdf | Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run. - CONFIRMED | X41: The nine numbers, and why there is no luckier polynomial | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; public/strata/ulam-spiral/index.html | REJECTED with evidence: the preceding paragraph defines lucky as a full run n=0..c-2. Granville and Mollin p.139 classifies that property, whose largest c>=2 is 41. The local heading/question is valid in that sense; an arbitrary-streak counterexample addresses another question. Left this wording intact. - STALE | X42: "dateModified":"2026-06-21" | git show 91d91ec04083e1d7a5e0415939709a2a25b9dbaf:public/strata/ulam-spiral/index.html; public/strata/ulam-spiral/index.html | The June publication-era date remained after a September 28 page correction. Set JSON-LD dateModified to 2026-10-14. The explicit metadata exception requires no separate Updated line; the substantive Corrected lines date this revision. - MINOR | X43: the full run of length c−1 = 1). Discriminant 1−4c = -7, a Heegner number | public/strata/ulam-spiral/index.html; https://math.nyu.edu/~tschinke/books/gauss-dirichlet/stark.pdf | Heegner numbers here are positive squarefree radicands. The shared lucky readout now calls D minus the positive Heegner number, and the table labels -D and displays the positive number. All six lucky buttons are covered; no new dated line. - MINOR | X44: the full run of length c−1 = 2). Discriminant 1−4c = -11, a Heegner number | public/strata/ulam-spiral/index.html; https://math.nyu.edu/~tschinke/books/gauss-dirichlet/stark.pdf | Heegner numbers here are positive squarefree radicands. The shared lucky readout now calls D minus the positive Heegner number, and the table labels -D and displays the positive number. All six lucky buttons are covered; no new dated line. - MINOR | X45: the full run of length c−1 = 4). Discriminant 1−4c = -19, a Heegner number | public/strata/ulam-spiral/index.html; https://math.nyu.edu/~tschinke/books/gauss-dirichlet/stark.pdf | Heegner numbers here are positive squarefree radicands. The shared lucky readout now calls D minus the positive Heegner number, and the table labels -D and displays the positive number. All six lucky buttons are covered; no new dated line. - MINOR | X46: the full run of length c−1 = 10). Discriminant 1−4c = -43, a Heegner number | public/strata/ulam-spiral/index.html; https://math.nyu.edu/~tschinke/books/gauss-dirichlet/stark.pdf | Heegner numbers here are positive squarefree radicands. The shared lucky readout now calls D minus the positive Heegner number, and the table labels -D and displays the positive number. All six lucky buttons are covered; no new dated line. - MINOR | X47: the full run of length c−1 = 16). Discriminant 1−4c = -67, a Heegner number | public/strata/ulam-spiral/index.html; https://math.nyu.edu/~tschinke/books/gauss-dirichlet/stark.pdf | Heegner numbers here are positive squarefree radicands. The shared lucky readout now calls D minus the positive Heegner number, and the table labels -D and displays the positive number. All six lucky buttons are covered; no new dated line. - MINOR | X48: the full run of length c−1 = 40). Discriminant 1−4c = -163, a Heegner number | public/strata/ulam-spiral/index.html; https://math.nyu.edu/~tschinke/books/gauss-dirichlet/stark.pdf | Heegner numbers here are positive squarefree radicands. The shared lucky readout now calls D minus the positive Heegner number, and the table labels -D and displays the positive number. All six lucky buttons are covered; no new dated line. - WRONG | X49: (added) the richest one owes its primes to a number called 163 | https://www.ams.org/journals/mcom/2003-72-241/S0025-5718-02-01418-7/S0025-5718-02-01418-7.pdf | Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run. - WRONG | X50: (added) Euler’s lucky c = 2, 3, 5, 11, 17, 41 are precisely the primes whose discriminant 1−4c — that is, −7, −11, −19, −43, −67, −163 — names a field of class number one. | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | Granville and Mollin p.139 requires squarefree 4c-1 and field class number one. At c=7, the field is Q(sqrt(-3)) with class number one but f(1)=9. Added squarefreeness and integer c>=2, including the full-run metaphor and six-constant characterization. - WRONG | K1: At a scientific meeting in 1963, sitting through what he later called “the presentation of a long and very boring paper,” Stanisław Ulam began to doodle. | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf | Gardner p.122 quotes only “long and very boring paper”. The added words are outside that quoted span in the source. Restored the span, with connective prose and comma outside the quotation. - WRONG | K2: 2, 3, 5, 7, 11, 13, no formula, no rhythm anyone has found in two thousand years of looking. | https://arxiv.org/abs/2004.01285 | Elsholtz’s abstract gives unconditional prime-producing formulas. Removed the blanket denial; the page now describes irregular prime gaps. - WRONG | K3: A horizontal or vertical line of the spiral is also a quadratic, but the diagonals are the ones whose values are always odd | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf; research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | Gardner’s checkerboard account and the actual walk give an even diagonal 2,12,30,56,90,132,182. Diagonals retain one parity; rows and columns alternate. Replaced the all-odd claim. - WRONG | K4: A random odd number near 1000 is prime about one time in seven | https://www.math.ucla.edu/~tao/preprints/Slides/austms.pdf; research/ulam-spiral/verify.mjs | Tao p.11 gives the odd-integer baseline 2/log N. Recomputed 2/log(1000)=0.289529654602 and counted 30 primes among the 100 odd integers 901..1099. Changed one in seven to about one in three and a half. - WRONG | K5: The polynomial n²+n+c is prime for all of n = 0 … c−2 if and only if the field ℚ(√(1−4c)) has class number 1. | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | Granville and Mollin p.139 requires squarefree 4c-1 and field class number one. At c=7, the field is Q(sqrt(-3)) with class number one but f(1)=9. Added squarefreeness and integer c>=2, including the full-run metaphor and six-constant characterization. - WRONG | K6: Euler’s lucky c = 2, 3, 5, 11, 17, 41 are precisely the primes whose discriminant 1−4c | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | Granville and Mollin p.139 requires squarefree 4c-1 and field class number one. At c=7, the field is Q(sqrt(-3)) with class number one but f(1)=9. Added squarefreeness and integer c>=2, including the full-run metaphor and six-constant characterization. - MINOR | K7: Kurt Heegner proved it in 1952 (and was disbelieved for a decade over a gap later shown to be reparable) | https://math.nyu.edu/~tschinke/books/gauss-dirichlet/stark.pdf | Stark p.250 describes continued rejection in 1963 and rehabilitation in 1967-1969 after the 1952 paper. Changed a decade to more than a decade; no new dated line. - WRONG | K8: it is provably the best one of its entire kind that can ever exist. | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; research/ulam-spiral/verify.mjs | Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - WRONG | K9: The forty-prime run is a world record that the structure of mathematics forbids anyone from breaking. | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; research/ulam-spiral/verify.mjs | Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - MINOR | K10: Every number on this page is checked; this is the one we mark as unknown, because that is what it is. | research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | The verifier performs finite arithmetic checks; the long near-integer decimal is a literal in the browser script. Narrowed prose to the listed checks and distinguished live calculations from supplied figures; no new dated line. The expanded arithmetic verifier passes 23/23. - WRONG | K11: why the opening run is exactly forty, why no parabola will ever beat it | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; research/ulam-spiral/verify.mjs | Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - MINOR | K12: Every claim above is recomputed offline in research/ulam-spiral/verify.mjs before the page asserts it, and the browser re-derives the live figures from scratch. | research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | The verifier performs finite arithmetic checks; the long near-integer decimal is a literal in the browser script. Narrowed prose to the listed checks and distinguished live calculations from supplied figures; no new dated line. The expanded arithmetic verifier passes 23/23. - MINOR | K13: The honesty bar: every number checked offline and re-derived live; the one open question named as open. | research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | The verifier performs finite arithmetic checks; the long near-integer decimal is a literal in the browser script. Narrowed prose to the listed checks and distinguished live calculations from supplied figures; no new dated line. The expanded arithmetic verifier passes 23/23. - WRONG | K14: The richest prime-making parabola anyone has found is Euler's n²+n+41 | https://www.ams.org/journals/mcom/2003-72-241/S0025-5718-02-01418-7/S0025-5718-02-01418-7.pdf | Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run. - WRONG | K15: Rabinowitsch's theorem (1913) says, with no slack, that the full run happens precisely when the imaginary quadratic field ℚ(√(1−4c)) has unique factorization, i.e. class number 1. | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | Granville and Mollin p.139 requires squarefree 4c-1 and field class number one. At c=7, the field is Q(sqrt(-3)) with class number one but f(1)=9. Added squarefreeness and integer c>=2, including the full-run metaphor and six-constant characterization. - WRONG | K16: So Euler's lucky numbers 2,3,5,11,17,41 are exactly the primes whose discriminants −7,−11,−19,−43,−67,−163 are the nine Heegner numbers | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; https://math.nyu.edu/~tschinke/books/gauss-dirichlet/stark.pdf; public/strata/ulam-spiral/index.html | The six constants map to discriminants -7,-11,-19,-43,-67,-163. The remaining class-number-one fields have discriminants -3,-4,-8. Corrected the identification: minus six of nine positive Heegner numbers, not the same census. - WRONG | K17: 163 is the last, so Euler's 40-prime run is a record the structure of mathematics forbids anyone to break | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; research/ulam-spiral/verify.mjs | Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - WRONG | K18: Each diagonal is a quadratic, and the richest anyone has found is Euler's n²+n+41, which yields 40 primes in a row. | https://www.ams.org/journals/mcom/2003-72-241/S0025-5718-02-01418-7/S0025-5718-02-01418-7.pdf | Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run. - WRONG | K19: A live spiral and a polynomial lab show, through Rabinowitsch's theorem and the Heegner numbers, why that record cannot be beaten. | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; research/ulam-spiral/verify.mjs | Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - WRONG | K20: it lays Euler's lucky numbers beside the nine Heegner numbers and shows them to be the same census | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; https://math.nyu.edu/~tschinke/books/gauss-dirichlet/stark.pdf; public/strata/ulam-spiral/index.html | The six constants map to discriminants -7,-11,-19,-43,-67,-163. The remaining class-number-one fields have discriminants -3,-4,-8. Corrected the identification: minus six of nine positive Heegner numbers, not the same census. - MINOR | K21: Every figure is recomputed offline in `research/ulam-spiral/verify.mjs` (17/17) before the page asserts it, and re-derived from scratch in your browser. | research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | The verifier performs finite arithmetic checks; the long near-integer decimal is a literal in the browser script. Narrowed prose to the listed checks and distinguished live calculations from supplied figures; no new dated line. The expanded arithmetic verifier passes 23/23. - WRONG | K22: the richest quadratic, Euler's n²+n+41, owes its 40 primes to the last of the nine Heegner numbers. | https://www.ams.org/journals/mcom/2003-72-241/S0025-5718-02-01418-7/S0025-5718-02-01418-7.pdf | Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run. - WRONG | K23: The most prime-rich parabola anyone has found | https://www.ams.org/journals/mcom/2003-72-241/S0025-5718-02-01418-7/S0025-5718-02-01418-7.pdf | Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run. - STALE | K24: "dateModified":"2026-06-21" | git show 91d91ec04083e1d7a5e0415939709a2a25b9dbaf:public/strata/ulam-spiral/index.html; public/strata/ulam-spiral/index.html | The June publication-era date remained after a September 28 page correction. Set JSON-LD dateModified to 2026-10-14. The explicit metadata exception requires no separate Updated line; the substantive Corrected lines date this revision. - WRONG | K25: (added) Ulam, 1963, at a dull lecture · ... · the richest one owes its primes to a number called 163 (subtitle, index.html line 114) | https://www.ams.org/journals/mcom/2003-72-241/S0025-5718-02-01418-7/S0025-5718-02-01418-7.pdf | Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run. - CONFIRMED | A mathematician doodles through a boring talk: 1, 2, 3, 4… wound into a square spiral. | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf | [opening-doodle] [001 opening-doodle] Gardner p.122 says "Last fall" in the March 1964 issue, "long and very boring paper", and "With no special end in view" before describing circling the primes. The supplied snippets are accurate; the surrounding account supplies the date and sequence. - CONFIRMED | He circles the primes. | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf | [circle-primes] [002 circle-primes] Gardner p.122 says "Last fall" in the March 1964 issue, "long and very boring paper", and "With no special end in view" before describing circling the primes. The supplied snippets are accurate; the surrounding account supplies the date and sequence. - CONFIRMED | They lie, unmistakably, along diagonal lines | https://studylib.net/doc/7950840/a-visual-display-of-some-properties-of-the-distribution-o... | [diagonal-lines] [003 diagonal-lines] Stein, Ulam and Wells p.517: "some other quadratic formsare rich in primes." The discussion describes the visibly nonrandom lines. Opened through the web reader after curl returned HTTP 403. - CONFIRMED | He wrote 1 in the middle of the page, then spiralled the counting numbers outward around it | https://studylib.net/doc/7950840/a-visual-display-of-some-properties-of-the-distribution-o... | [ulam-start] [005 ulam-start] The supplied leading-coefficient excerpt does not establish the starting point or orientation. The same paper p.516 does: "(0, 0)-1, (1, O)->2, (1, 1)->3, (O, 1)->4, (-1, 1)-?5" in the web transcription. Its prose says "startingat (0, 0) and proceedingcounterclockwise". The page and independent spiral walk agree. - CONFIRMED | 2 to the right, 3 above that, 4 and 5 to the left, and on, square ring after square ring. | https://studylib.net/doc/7950840/a-visual-display-of-some-properties-of-the-distribution-o... | [ulam-order] [006 ulam-order] The supplied leading-coefficient excerpt does not establish the starting point or orientation. The same paper p.516 does: "(0, 0)-1, (1, O)->2, (1, 1)->3, (O, 1)->4, (-1, 1)-?5" in the web transcription. Its prose says "startingat (0, 0) and proceedingcounterclockwise". The page and independent spiral walk agree. - CONFIRMED | Then he circled the primes. | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf | [ulam-circles] [007 ulam-circles] Gardner p.122 says "Last fall" in the March 1964 issue, "long and very boring paper", and "With no special end in view" before describing circling the primes. The supplied snippets are accurate; the surrounding account supplies the date and sequence. - CONFIRMED | He expected nothing. | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf | [ulam-expectation] [008 ulam-expectation] Gardner p.122 says "Last fall" in the March 1964 issue, "long and very boring paper", and "With no special end in view" before describing circling the primes. The supplied snippets are accurate; the surrounding account supplies the date and sequence. - CONFIRMED | But circled on the spiral they did something startling: they lined up on the diagonals. | https://studylib.net/doc/7950840/a-visual-display-of-some-properties-of-the-distribution-o... | [doodle-diagonals] [011 doodle-diagonals] Stein, Ulam and Wells p.517: "some other quadratic formsare rich in primes." The discussion describes the visibly nonrandom lines. Opened through the web reader after curl returned HTTP 403. - CONFIRMED | Whole diagonal streets ran thick with primes; others stayed nearly empty. | https://studylib.net/doc/7950840/a-visual-display-of-some-properties-of-the-distribution-o... | [unequal-lines] [012 unequal-lines] Stein, Ulam and Wells p.517: "some other quadratic formsare rich in primes." The discussion describes the visibly nonrandom lines. Opened through the web reader after curl returned HTTP 403. - CONFIRMED | Martin Gardner put the picture on the cover of Scientific American in March 1964 | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf | [gardner-cover] [014 gardner-cover] Gardner p.122: "The illustration on the cover of this issue shows how the primes appeared on the spiral grid from 1 to 100." The issue and publisher page identify March 1964. - CONFIRMED | The southwest diagonal is 4n²+2n+1; the northeast is 4n²−2n+1; the southeast holds the odd squares (2n+1)². | https://studylib.net/doc/7950840/a-visual-display-of-some-properties-of-the-distribution-o...; git show 67088387e0:research/ulam-spiral/verify.mjs | [three-formulas] [019 three-formulas] Independent coordinate walking and the verifier give SW values "1,7,21,43,73", NE "1,3,13,31,57", SE "1,9,25,49,81". Their formulas are the three quoted ones for n >= 0. The record's generic leading-term excerpt alone does not fix these directions. - CONFIRMED | an even number bigger than two is never prime, so half the board is dead on arrival. | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf | [even-exception] [022 even-exception] Gardner p.122: "all primes except 2 are odd." Its checkerboard experiment explains why odd occupied squares form diagonal lines. The parity reasoning is correct. - CONFIRMED | So some diagonal preference is forced, and unmysterious. | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf | [forced-preference] [024 forced-preference] Gardner p.122: "all primes except 2 are odd." Its checkerboard experiment explains why odd occupied squares form diagonal lines. The parity reasoning is correct. - CONFIRMED | Two parabolas, both all-odd, can differ wildly in how many primes they catch. | https://www.researchgate.net/publication/2807438_New_quadratic_polynomials_with_high_densities_of_prime_values | [odd-parabolas] [025 odd-parabolas] Jacobson and Williams p.500 report polynomials with "higher asymptotic densities of prime values" than Euler's. Both x²+x+41 and their larger odd-constant example are always odd. The claim of differing prime richness is supported. - CONFIRMED | That is the real question Ulam’s doodle asks, and it turns out to be the same question Euler asked in 1772 | https://repository.kulib.kyoto-u.ac.jp/bitstream/2433/196238/1/B32_016.pdf; https://scholarlycommons.pacific.edu/euler-works/461/ | [euler-question] [026 euler-question] The supplied archive excerpt is only the volume label "Volume 1772"; that record separately gives "Published Date 1774" and "Written Date 1774". Supplementary mathematical research in RIMS Kokyuroku Bessatsu B32, section 2, explicitly says "In 1772, L.Euler discovered that the quadratic polynomial" x²+x+41 has the stated prime values. The historical year is corroborated by that search-indexed research passage, but not by the excerpt N1 selected. A fresh direct fetch of the Kyoto URL returned a repository HTML shell, so the supplemental PDF was not freshly readable. - CONFIRMED | and the same question that is still, in its general form, unsolved. | https://kconrad.math.uconn.edu/blurbs/ugradnumthy/prime-patterns-1.pdf | [general-open] [027 general-open] Conrad, Patterns in Primes, p.3: "There is no case of Bunyakovsky’s conjecture that is proved when deg f ≥ 2." This includes the irreducible quadratic x²+x+41. The source supports the named open problem. - CONFIRMED | In 1772 Leonhard Euler noticed something about the humble polynomial n² + n + 41. | https://repository.kulib.kyoto-u.ac.jp/bitstream/2433/196238/1/B32_016.pdf; https://scholarlycommons.pacific.edu/euler-works/461/ | [euler-1772] [028 euler-1772] The supplied archive excerpt is only the volume label "Volume 1772"; that record separately gives "Published Date 1774" and "Written Date 1774". Supplementary mathematical research in RIMS Kokyuroku Bessatsu B32, section 2, explicitly says "In 1772, L.Euler discovered that the quadratic polynomial" x²+x+41 has the stated prime values. The historical year is corroborated by that search-indexed research passage, but not by the excerpt N1 selected. A fresh direct fetch of the Kyoto URL returned a repository HTML shell, so the supplemental PDF was not freshly readable. - CONFIRMED | Feed it n = 0 and you get 41, a prime. | https://scholarlycommons.pacific.edu/euler-works/461/; git show 67088387e0:research/ulam-spiral/verify.mjs | [euler-zero] [029 euler-zero] Euler Archive: "41–x+xx produces prime numbers for every integer value of x from 0 to 40." Set x=n+1 to obtain n²+n+41. Independently checked values 41,43,47,53,61,71,83,97, all n=0..39 prime, f(39)=1601, and f(40)=1681=41². The archive excerpt by itself does not state the composite endpoint; arithmetic supplies it. - CONFIRMED | n = 1 gives 43, prime. | https://scholarlycommons.pacific.edu/euler-works/461/; git show 67088387e0:research/ulam-spiral/verify.mjs | [euler-one] [030 euler-one] Euler Archive: "41–x+xx produces prime numbers for every integer value of x from 0 to 40." Set x=n+1 to obtain n²+n+41. Independently checked values 41,43,47,53,61,71,83,97, all n=0..39 prime, f(39)=1601, and f(40)=1681=41². The archive excerpt by itself does not state the composite endpoint; arithmetic supplies it. - CONFIRMED | Then 47, 53, 61, 71, 83, 97… | https://scholarlycommons.pacific.edu/euler-works/461/; git show 67088387e0:research/ulam-spiral/verify.mjs | [euler-next] [031 euler-next] Euler Archive: "41–x+xx produces prime numbers for every integer value of x from 0 to 40." Set x=n+1 to obtain n²+n+41. Independently checked values 41,43,47,53,61,71,83,97, all n=0..39 prime, f(39)=1601, and f(40)=1681=41². The archive excerpt by itself does not state the composite endpoint; arithmetic supplies it. - CONFIRMED | not for a while, but for forty values in a row, n = 0 through 39, ending at 39²+39+41 = 1601. | https://scholarlycommons.pacific.edu/euler-works/461/; git show 67088387e0:research/ulam-spiral/verify.mjs | [euler-forty] [032 euler-forty] Euler Archive: "41–x+xx produces prime numbers for every integer value of x from 0 to 40." Set x=n+1 to obtain n²+n+41. Independently checked values 41,43,47,53,61,71,83,97, all n=0..39 prime, f(39)=1601, and f(40)=1681=41². The archive excerpt by itself does not state the composite endpoint; arithmetic supplies it. - CONFIRMED | Only at n = 40 does it finally fail, and it fails for a reason you can see at a glance: 40²+40+41 = 1681 = 41². | https://scholarlycommons.pacific.edu/euler-works/461/; git show 67088387e0:research/ulam-spiral/verify.mjs | [euler-failure] [033 euler-failure] Euler Archive: "41–x+xx produces prime numbers for every integer value of x from 0 to 40." Set x=n+1 to obtain n²+n+41. Independently checked values 41,43,47,53,61,71,83,97, all n=0..39 prime, f(39)=1601, and f(40)=1681=41². The archive excerpt by itself does not state the composite endpoint; arithmetic supplies it. - CONFIRMED | The streak had to die exactly there, because 40²+40+41 = 41·(40+1) | https://scholarlycommons.pacific.edu/euler-works/461/; git show 67088387e0:research/ulam-spiral/verify.mjs | [euler-divisibility] [034 euler-divisibility] Euler Archive: "41–x+xx produces prime numbers for every integer value of x from 0 to 40." Set x=n+1 to obtain n²+n+41. Independently checked values 41,43,47,53,61,71,83,97, all n=0..39 prime, f(39)=1601, and f(40)=1681=41². The archive excerpt by itself does not state the composite endpoint; arithmetic supplies it. - CONFIRMED | once n = 40, the 41 divides it. | https://scholarlycommons.pacific.edu/euler-works/461/; git show 67088387e0:research/ulam-spiral/verify.mjs | [euler-factor] [035 euler-factor] Euler Archive: "41–x+xx produces prime numbers for every integer value of x from 0 to 40." Set x=n+1 to obtain n²+n+41. Independently checked values 41,43,47,53,61,71,83,97, all n=0..39 prime, f(39)=1601, and f(40)=1681=41². The archive excerpt by itself does not state the composite endpoint; arithmetic supplies it. - CONFIRMED | Euler’s polynomial hits forty for forty. | https://scholarlycommons.pacific.edu/euler-works/461/; git show 67088387e0:research/ulam-spiral/verify.mjs | [forty-for-forty] [037 forty-for-forty] Euler Archive: "41–x+xx produces prime numbers for every integer value of x from 0 to 40." Set x=n+1 to obtain n²+n+41. Independently checked values 41,43,47,53,61,71,83,97, all n=0..39 prime, f(39)=1601, and f(40)=1681=41². The archive excerpt by itself does not state the composite endpoint; arithmetic supplies it. - CONFIRMED | Drop it onto a spiral and it would be one of those blazing diagonals. | https://studylib.net/doc/7950840/a-visual-display-of-some-properties-of-the-distribution-o... | [euler-on-spiral] [039 euler-on-spiral] Stein p.517 explicitly relates the prominent spiral polynomial 4x²+170x+1847 to Euler's form by "putting y=2x+42." Thus a subsequence of Euler values does occur on a bright spiral line. This is not literally a unit-step diagonal with leading coefficient 1; the prose says "would be". - CONFIRMED | 41 is the largest constant c giving the full c-1 opening run for n²+n+c. | https://www.impan.pl/shop/en/publication/transaction/download/product/82930 | [champion-family] Confirmed at the full-run scope defined on the page, with the unrestricted champion wording narrowed alongside X14. - CONFIRMED | every value from n=0 to n=c−2 prime | https://www.impan.pl/shop/en/publication/transaction/download/product/82930 | [full-run-definition] [042 full-run-definition] Granville and Mollin p.139 require "4A − 1 is squarefree" as well as class number one, and give "A = 1, 2, 3, 5, 11, 17 and 41." For prime c >= 2 the six full c−1 runs are exactly the stated list. The page's footer correctly includes the squarefree/fundamental distinction. - CONFIRMED | happens only for c = 2, 3, 5, 11, 17, 41. | https://www.impan.pl/shop/en/publication/transaction/download/product/82930 | [lucky-list] [043 lucky-list] Granville and Mollin p.139 require "4A − 1 is squarefree" as well as class number one, and give "A = 1, 2, 3, 5, 11, 17 and 41." For prime c >= 2 the six full c−1 runs are exactly the stated list. The page's footer correctly includes the squarefree/fundamental distinction. - CONFIRMED | These are Euler’s lucky numbers, and there are exactly six of them. | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1912.1/ICM1912.1.ocr.pdf | [six-lucky] [044 six-lucky] The six values are correct, but the selected 1913 primary source does not prove exhaustiveness. Immediately after the list it asks "Gibt es noch solche Primzahlen" and calls the question "ungelöst". Granville and Mollin p.139 provide the later classification and support "exactly six". - CONFIRMED | Nothing larger ever works again. | https://www.impan.pl/shop/en/publication/transaction/download/product/82930 | [nothing-larger] [045 nothing-larger] Granville and Mollin p.139 require "4A − 1 is squarefree" as well as class number one, and give "A = 1, 2, 3, 5, 11, 17 and 41." For prime c >= 2 the six full c−1 runs are exactly the stated list. The page's footer correctly includes the squarefree/fundamental distinction. - CONFIRMED | To a quadratic like n²+n+c you can attach a single number, its discriminant 1 − 4c | https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1912.1/ICM1912.1.ocr.pdf | [discriminant] [047 discriminant] Rabinovitch p.418 writes "D = 1 - 4m" for these quadratic fields. Direct discriminant calculation is b²−4ac=1−4c; c=41 gives −163. Adjoining its square root to Q is precisely the stated imaginary quadratic field. - CONFIRMED | for Euler’s polynomial, 1 − 164 = −163. | https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1912.1/ICM1912.1.ocr.pdf | [euler-discriminant] [048 euler-discriminant] Rabinovitch p.418 writes "D = 1 - 4m" for these quadratic fields. Direct discriminant calculation is b²−4ac=1−4c; c=41 gives −163. Adjoining its square root to Q is precisely the stated imaginary quadratic field. - CONFIRMED | That negative number names an imaginary quadratic field, the number system you get by adjoining √(1−4c) to the ordinary fractions. | https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1912.1/ICM1912.1.ocr.pdf | [imaginary-field] [049 imaginary-field] Rabinovitch p.418 writes "D = 1 - 4m" for these quadratic fields. Direct discriminant calculation is b²−4ac=1−4c; c=41 gives −163. Adjoining its square root to Q is precisely the stated imaginary quadratic field. - CONFIRMED | A field’s class number measures the failure of unique factorization in its ring of integers. | https://www.impan.pl/shop/en/publication/transaction/download/product/82930 | [class-number-meaning] The ring-of-integers scope is now explicit in the same paragraph as the corrected criterion. - CONFIRMED | how far the system is from a world where every number factors into primes in one and only one way. | https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1912.1/ICM1912.1.ocr.pdf | [unique-factorization] [051 unique-factorization] Rabinovitch p.418 asks whether decomposition into prime factors is unique and says the question is equivalent to "ist die Klassenzahl eines gegebenen Körpers gleich 1 oder grösser ?" Read with the usual convention: factorization of algebraic integers, up to units and order, not arbitrary field elements. - CONFIRMED | Class number 1 means factorization is perfectly unique, just like the ordinary integers; bigger class numbers mean it breaks. | https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1912.1/ICM1912.1.ocr.pdf | [class-number-one] [052 class-number-one] Rabinovitch p.418 asks whether decomposition into prime factors is unique and says the question is equivalent to "ist die Klassenzahl eines gegebenen Körpers gleich 1 oder grösser ?" Read with the usual convention: factorization of algebraic integers, up to units and order, not arbitrary field elements. - CONFIRMED | that is, −7, −11, −19, −43, −67, −163 | https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1912.1/ICM1912.1.ocr.pdf | [six-discriminant-values] [057 six-discriminant-values] Direct substitution c=2,3,5,11,17,41 gives −7,−11,−19,−43,−67,−163, exactly the discriminants printed on Rabinovitch p.421. - CONFIRMED | 1, 2, 3, 7, 11, 19, 43, 67, 163 | https://www.impan.pl/shop/en/publication/transaction/download/product/82930 | [nine-values] [060 nine-values] Granville and Mollin p.139 list A=1,2,3,5,11,17,41 and add Q(√−4), Q(√−8) in footnote 1. Removing square factors gives exactly the nine positive squarefree radicands printed on the page. - CONFIRMED | Carl Friedrich Gauss conjectured the list around 1801 | https://math.nyu.edu/~tschinke/books/gauss-dirichlet/stark.pdf | [gauss-1801] [061 gauss-1801] Stark p.247: "In Articles 303 and 304 of his 1801 Disquisitiones Arithmeticae" Gauss put forward the conjectures; he surmised his low-class-number tables were complete. - CONFIRMED | Alan Baker and Harold Stark independently nailed it shut in 1966–67. | https://math.nyu.edu/~tschinke/books/gauss-dirichlet/stark.pdf | [baker-stark] [063 baker-stark] Stark p.249 says the problem "was solved by Baker [Bak66] and Stark [Sta67] completely" and discusses "the earlier discounted method of Heegner [Hee52] from 1952" that could be made valid. He recounts renewed rejection in 1963. His p.251 states "there are no class-number one fields past −163". The dates and historical summary check out. - CONFIRMED | 163 is the last and largest. | https://math.nyu.edu/~tschinke/books/gauss-dirichlet/stark.pdf | [largest-163] [064 largest-163] Stark p.249 says the problem "was solved by Baker [Bak66] and Stark [Sta67] completely" and discusses "the earlier discounted method of Heegner [Hee52] from 1952" that could be made valid. He recounts renewed rejection in 1963. His p.251 states "there are no class-number one fields past −163". The dates and historical summary check out. - CONFIRMED | The same 163 is the reason behind a second, gaudier coincidence: eπ√163 is almost exactly a whole number, 262 537 412 640 768 744 minus about 0.00000000000075 | https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/Mysore-12-2003.pdf | [near-integer] [067 near-integer] Waldschmidt slides 2 and 7 give "262537412640768743.99999999999925" and state the relevant imaginary quadratic field "has class number 1." Slide 7 links this to an integral modular-function value. Independent 70-digit mpmath arithmetic agrees. The three-faces sentence is read as the mathematical connection, not as claiming class numbers explain every diagonal. - CONFIRMED | close enough that it fooled people into thinking it was an integer, and earned the nickname Ramanujan’s constant. | https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/Mysore-12-2003.pdf | [constant-nickname] [068 constant-nickname] Waldschmidt slide 3 attributes the name to "an April Fool’s joke played by Martin Gardner (Apr. 1975)" and records Gardner's integer claim and later admission. It supports the familiar nickname/hoax account. - CONFIRMED | That near-miss is class number one again, viewed through a different window. | https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/Mysore-12-2003.pdf | [constant-class-number] [070 constant-class-number] Waldschmidt slides 2 and 7 give "262537412640768743.99999999999925" and state the relevant imaginary quadratic field "has class number 1." Slide 7 links this to an integral modular-function value. Independent 70-digit mpmath arithmetic agrees. The three-faces sentence is read as the mathematical connection, not as claiming class numbers explain every diagonal. - CONFIRMED | The bored doodle, the forty primes, and the almost-integer are three faces of a single deep fact. | https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/Mysore-12-2003.pdf | [three-faces] [071 three-faces] Waldschmidt slides 2 and 7 give "262537412640768743.99999999999925" and state the relevant imaginary quadratic field "has class number 1." Slide 7 links this to an integral modular-function value. Independent 70-digit mpmath arithmetic agrees. The three-faces sentence is read as the mathematical connection, not as claiming class numbers explain every diagonal. - CONFIRMED | In 1923 G. H. Hardy and J. E. Littlewood wrote down a precise prediction | https://www.researchgate.net/publication/2807438_New_quadratic_polynomials_with_high_densities_of_prime_values; https://api.crossref.org/works/10.1007/BF02403921 | [hl-1923] [073 hl-1923] Jacobson and Williams p.499 name "Conjecture (F)" and credit Hardy and Littlewood; their reference [10] and Crossref identify the 1923 Acta Mathematica paper. The page correctly labels this a prediction. - CONFIRMED | For Euler’s polynomial it forecasts a prime crop about 6.6 times richer than random integers of the same size predict | https://www.researchgate.net/publication/2807438_New_quadratic_polynomials_with_high_densities_of_prime_values | [density-prediction] [075 density-prediction] Jacobson and Williams p.500 give "C(−163) = 3.3197732" and normalize by twice the logarithmic integral of 1/log(f(x)). Thus the factor against random integers is 6.6395464; against odd integers it is 3.3197732. These are predictions, as the page states. - CONFIRMED | Conjecture F has never been proved. | https://kconrad.math.uconn.edu/blurbs/ugradnumthy/prime-patterns-1.pdf | [hl-unproved] [077 hl-unproved] Conrad, Patterns in Primes, p.3: "There is no case of Bunyakovsky’s conjecture that is proved when deg f ≥ 2." This includes the irreducible quadratic x²+x+41. The source supports the named open problem. - CONFIRMED | Worse: it is not even known whether n²+n+41 produces infinitely many primes at all | https://kconrad.math.uconn.edu/blurbs/ugradnumthy/prime-patterns-1.pdf | [euler-infinitely-often] [078 euler-infinitely-often] Conrad, Patterns in Primes, p.3: "There is no case of Bunyakovsky’s conjecture that is proved when deg f ≥ 2." This includes the irreducible quadratic x²+x+41. The source supports the named open problem. - CONFIRMED | no quadratic polynomial has ever been proved to be prime infinitely often. | https://kconrad.math.uconn.edu/blurbs/ugradnumthy/prime-patterns-1.pdf | [no-quadratic-proof] [079 no-quadratic-proof] Conrad, Patterns in Primes, p.3: "There is no case of Bunyakovsky’s conjecture that is proved when deg f ≥ 2." This includes the irreducible quadratic x²+x+41. The source supports the named open problem. - CONFIRMED | The diagonals on Ulam’s spiral are real, and we can predict their brightness to several decimals, but we cannot prove the brightest one never goes dark. | https://www.researchgate.net/publication/2807438_New_quadratic_polynomials_with_high_densities_of_prime_values | [predict-decimals] [080 predict-decimals] Jacobson and Williams p.500 give "C(−163) = 3.3197732" and normalize by twice the logarithmic integral of 1/log(f(x)). Thus the factor against random integers is 6.6395464; against odd integers it is 3.3197732. These are predictions, as the page states. - CONFIRMED | The six lucky constants have full c-1 runs, and 1-4c is minus six of the nine Heegner numbers. | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | [lucky-exact-run] Each has squarefree discriminant and field class number one; the verifier checks the six exact runs. - CONFIRMED | the proof that there is none at all, ever, is Heegner 1952 / Baker & Stark 1966–67. | https://math.nyu.edu/~tschinke/books/gauss-dirichlet/stark.pdf | [no-tenth-proof] [089 no-tenth-proof] Stark p.249 says the problem "was solved by Baker [Bak66] and Stark [Sta67] completely" and discusses "the earlier discounted method of Heegner [Hee52] from 1952" that could be made valid. He recounts renewed rejection in 1963. His p.251 states "there are no class-number one fields past −163". The dates and historical summary check out. - CONFIRMED | Hardy–Littlewood Conjecture F (1923) predicts the prime density | https://www.researchgate.net/publication/2807438_New_quadratic_polynomials_with_high_densities_of_prime_values; https://api.crossref.org/works/10.1007/BF02403921 | [hl-density-footer] [090 hl-density-footer] Jacobson and Williams p.499 name "Conjecture (F)" and credit Hardy and Littlewood; their reference [10] and Crossref identify the 1923 Acta Mathematica paper. The page correctly labels this a prediction. - CONFIRMED | the Hardy–Littlewood constant, taken against odd integers, is half of that, ≈3.32) | https://www.researchgate.net/publication/2807438_New_quadratic_polynomials_with_high_densities_of_prime_values | [half-constant] [092 half-constant] Jacobson and Williams p.500 give "C(−163) = 3.3197732" and normalize by twice the logarithmic integral of 1/log(f(x)). Thus the factor against random integers is 6.6395464; against odd integers it is 3.3197732. These are predictions, as the page states. - CONFIRMED | The conjecture is unproven, and it is not known that any quadratic is prime infinitely often | https://kconrad.math.uconn.edu/blurbs/ugradnumthy/prime-patterns-1.pdf | [open-footer] [093 open-footer] Conrad, Patterns in Primes, p.3: "There is no case of Bunyakovsky’s conjecture that is proved when deg f ≥ 2." This includes the irreducible quadratic x²+x+41. The source supports the named open problem. - CONFIRMED | M. L. Stein, S. M. Ulam & M. B. Wells, “A Visual Display of Some Properties of the Distribution of Primes,” American Mathematical Monthly 71 (1964) 516–520. | https://api.crossref.org/works/10.2307/2312588; https://studylib.net/doc/7950840/a-visual-display-of-some-properties-of-the-distribution-o... | [citation-1] [099 citation-1] Crossref names "M. L. Stein", "S. M. Ulam", "M. B. Wells", the exact title, volume 71 and May 1964. The primary article itself runs pp.516-520. Crossref alone only supplies first page "516". - CONFIRMED | M. Gardner, “Mathematical Games,” Scientific American 210 no. 3 (March 1964). | https://www.scientificamerican.com/article/mathematical-games-1964-03/ | [citation-2] [100 citation-2] Publisher page states "By Martin Gardner" and "Scientific American Magazine Vol. 210 No. 3 (March 1964), p. 120", title "Mathematical Games". DOI 10.1038/scientificamerican0364-120 matches. - CONFIRMED | L. Euler, on x²+x+41 (1772). | https://scholarlycommons.pacific.edu/euler-works/461/ | [citation-3] [101 citation-3] The citation's 1772 is the nominal volume year: "Volume 1772, pp. 35-36." The Euler Archive separately records publication in 1774. This is acceptable as a historical volume citation but should not be used alone to date discovery. The original polynomial is x²−x+41, transformed by x=n+1. - CONFIRMED | G. Rabinowitsch, “Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern,” Proc. Fifth Int. Congress of Mathematicians, Cambridge (1913). | https://www.impan.pl/shop/en/publication/transaction/download/product/82930; https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1912.1/ICM1912.1.ocr.pdf | [citation-4] [102 citation-4] Granville and Mollin reference [22] supplies the quoted title, G. Rabinowitsch, volume 1, "1913, 418–421". The original proceedings uses the spelling "G. RABINOVITCH". 1913 is the publication year, not the congress year. - CONFIRMED | K. Heegner (1952); A. Baker (1966); H. M. Stark (1967), on the class-number-one problem. | https://math.nyu.edu/~tschinke/books/gauss-dirichlet/stark.pdf | [citation-5] [103 citation-5] Stark p.249 says the problem "was solved by Baker [Bak66] and Stark [Sta67] completely" and discusses "the earlier discounted method of Heegner [Hee52] from 1952" that could be made valid. He recounts renewed rejection in 1963. His p.251 states "there are no class-number one fields past −163". The dates and historical summary check out. - CONFIRMED | G. H. Hardy & J. E. Littlewood, “Some problems of ‘Partitio Numerorum’ III,” Acta Mathematica 44 (1923), Conjecture F. | https://api.crossref.org/works/10.1007/BF02403921; https://www.researchgate.net/publication/2807438_New_quadratic_polynomials_with_high_densities_of_prime_values | [citation-6] [104 citation-6] Crossref returns the title "Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes", authors G. H. Hardy and J. E. Littlewood, year 1923, volume 44, pages 1-70. Jacobson and Williams identify Conjecture F. - CONFIRMED | In 1963, bored at a lecture, Ulam wrote the whole numbers in a square spiral and circled the primes, and they fell onto diagonal lines. | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf | [dek-history] [113 dek-history] Gardner p.122 says "Last fall" in the March 1964 issue, "long and very boring paper", and "With no special end in view" before describing circling the primes. The supplied snippets are accurate; the surrounding account supplies the date and sequence. - CONFIRMED | An immersive layer; it renders as its own standalone page at the `href` above | git show 67088387e0:src/content/strata/ulam-spiral.md; https://artwaste.land/strata/ulam-spiral/ | [md-standalone] [126 md-standalone] Pinned Markdown has "href: "/strata/ulam-spiral/"". The standalone HTML supplies the complete page. Exact body phrase "An immersive layer; it renders as its own standalone page" is absent from the served HTML, while its title "The Doodle That Sees the Primes" is a positive control. The live HTML matches the pinned file except injected analytics. - CONFIRMED | its body is not rendered. | git show 67088387e0:src/content/strata/ulam-spiral.md; https://artwaste.land/strata/ulam-spiral/ | [md-unrendered] [127 md-unrendered] Pinned Markdown has "href: "/strata/ulam-spiral/"". The standalone HTML supplies the complete page. Exact body phrase "An immersive layer; it renders as its own standalone page" is absent from the served HTML, while its title "The Doodle That Sees the Primes" is a positive control. The live HTML matches the pinned file except injected analytics. - CONFIRMED | the zeros of the zeta function rebuild the primes as a chord. | git show 67088387e0:src/content/strata/harmonics-of-the-primes.md; https://kconrad.math.uconn.edu/blurbs/ugradnumthy/prime-patterns-1.pdf | [rel-harmonics] [139 rel-harmonics] Pinned companion content names "Riemann–von Mangoldt explicit formula", describes reconstructing the prime staircase from zeta zeros, and says the Riemann Hypothesis "remains unproven". This supports the description of that layer. Conrad separately supports the quadratic open problem. - CONFIRMED | Both end at the same honest wall: the spectral side names the Riemann Hypothesis as unproven; the spiral side names Hardy–Littlewood Conjecture F, and the open question of whether any quadratic is prime infinitely often. | git show 67088387e0:src/content/strata/harmonics-of-the-primes.md; https://kconrad.math.uconn.edu/blurbs/ugradnumthy/prime-patterns-1.pdf | [rel-open] [140 rel-open] Pinned companion content names "Riemann–von Mangoldt explicit formula", describes reconstructing the prime staircase from zeta zeros, and says the Riemann Hypothesis "remains unproven". This supports the description of that layer. Conrad separately supports the quadratic open problem. - CONFIRMED | There Gilbreath's leading-1 is really about gap structure, not primality | git show 67088387e0:src/content/strata/one-all-the-way-down.md | [rel-gilbreath] [141 rel-gilbreath] Pinned companion says "it is not special to the primes" and describes Croft's control experiment and the {0,2} fixed point. This checks the cross-reference's characterization, not a claim that Gilbreath's conjecture has been proved. - CONFIRMED | here the spiral's diagonals are really about quadratic discriminants and class numbers | git show 67088387e0:public/strata/ulam-spiral/index.html | [rel-spiral-structure] [142 rel-spiral-structure] The page actually renders primes by spiral coordinates and connects selected quadratic discriminants to class numbers, while section VI says "They do not" explain the whole spiral. These cross-reference descriptions are supported in that limited sense. - CONFIRMED | There the Lucas–Lehmer map x²−2 turns out to be angle-doubling | git show 67088387e0:src/content/strata/square-minus-two.md | [rel-ll] [143 rel-ll] Pinned companion gives "x = y + 1/y" and "x² − 2 = y² + 1/y²". Taking y=exp(iθ) produces 2cos(2θ) from 2cos(θ), the advertised doubling identity. - CONFIRMED | The Ulam spiral makes local prime structure visible in space. | git show 67088387e0:public/strata/ulam-spiral/index.html | [nearby-local] [144 nearby-local] The page actually renders primes by spiral coordinates and connects selected quadratic discriminants to class numbers, while section VI says "They do not" explain the whole spiral. These cross-reference descriptions are supported in that limited sense. - CONFIRMED | They are not scattered. | https://studylib.net/doc/7950840/a-visual-display-of-some-properties-of-the-distribution-o... | [not-scattered] [145 not-scattered] Stein, Ulam and Wells p.517: "some other quadratic formsare rich in primes." The discussion describes the visibly nonrandom lines. Opened through the web reader after curl returned HTTP 403. - CONFIRMED | (added) Two spirals that turn out to be about a hidden constant law. | git show 67088387e0:src/content/strata/the-chase-that-comes-to-a-point.md | [added-nearby-pursuit] [195 added-nearby-pursuit] The companion's pinned note says "every bug crosses every spoke from the center at one fixed angle". Its body gives the constant lead angle "90 - 180/n". The Ulam page gives constant quadratic coefficients on the central outward rays. The cross-reference characterization is supported. Exact phrase search in N1 found no entry, while "The Ulam spiral makes local prime structure visible in space." was the positive control. - CONFIRMED | Primes on the number line have irregular gaps. | research/ulam-spiral/verify.mjs | [no-pattern] The old generic “no pattern” prose was replaced with X2. The consecutive displayed primes 2,3,5,7,11,13 have gaps 1,2,2,4,2. - UNVERIFIABLE | Ulam went home, had the spiral plotted by computer out to tens of thousands, and the diagonals only sharpened. | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf (Gardner, Mathematical Games, Sci. Am. March 1964, p.122) | [computer-history] Checker disagreement retained: Gardner p.122 supports a MANIAC plot made with Stein and Wells, out to about 65,000. It does not establish “went home”. Codex marked this unsupported while Claude confirmed the combined sentence. The phrase is outside the triage and remains unedited; not promoted to CONFIRMED here. - CONFIRMED | Here is the spiral, drawn live in your browser. | public/strata/ulam-spiral/index.html at 67088387e0 | [live-spiral] [live-spiral] Page script draws the spiral on canvas#spiral at load (draw()). - CONFIRMED | Every prime up to 40,401 is a dark cell; the composites are left blank. | public/strata/ulam-spiral/index.html at 67088387e0 | [cells-40401] [cells-40401] "const SIDE=201; // odd; 201x201 = 40401 cells"; sieve to NMAX; composites skipped. - CONFIRMED | The four central outward rays fit 4n²+bn+c for integers b,c and n>=0. | research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | [fit-controls] The rays start at the centre. The browser fits three values and checks the next three; the offline verifier checks 60 terms on each ray. Negative b is allowed. - CONFIRMED | Across the first ten thousand values it is prime 41% of the time. | research/ulam-spiral/verify.mjs at 67088387e0 (recreated: node verify.mjs, 17/17) | [ten-thousand-density] [ten-thousand-density] Recreated (sympy): 4149 primes among n = 0..9999 (41.5%); verifier: "yields primes for 4149 of n=0..10000 (~41%)". - CONFIRMED | Pick the constant c in n²+n+c and the lab below runs the polynomial, lights every prime it produces, and tells you how long the opening run of primes lasts. | public/strata/ulam-spiral/index.html at 67088387e0 | [lab-function] [lab-function] runLab(): drawComb lights primes; "opening run" computed. - CONFIRMED | (The gap is far too small for the ordinary 64-bit arithmetic in this page to even see; the value below is what your browser computes, rounded, and it differs from the true value by hundreds | public/strata/ulam-spiral/index.html at 67088387e0 | [browser-precision] [browser-precision] Recreated in node: Math.exp(Math.PI*Math.sqrt(163)).toPrecision(18) = 262537412640768256, 488 below the integer. - CONFIRMED | The four central outward rays follow 4n²+4n+1, 4n²+1, 4n²-2n+1, 4n²+2n+1 for n>=0. | research/ulam-spiral/verify.mjs | [four-diagonals-checked] All four checks pass on 60 terms each; no whole-offset-diagonal assertion is inferred. - CONFIRMED | Field class numbers are counted at fundamental discriminants and agree with h(-15)=2, h(-23)=3, h(-47)=5, h(-71)=7, h(-163)=1. | research/ulam-spiral/verify.mjs; public/strata/ulam-spiral/index.html | [forms-calibration] The expanded verifier passes the existing calibration and added order-versus-field counterexamples. - CONFIRMED | The nine Heegner fields have h=1; no additional positive squarefree d<=100000 gives h=1. | research/ulam-spiral/verify.mjs | [nine-verified] The exhaustive finite check passed. Completeness beyond the bound is attributed to the theorem, not this computation. - CONFIRMED | the diagonal readout said each diagonal "is exactly" its quadratic after fitting only three values, which three points always fit. | git show 91d91ec0408 -- public/strata/ulam-spiral/index.html | [old-readout] [old-readout] Removed lines: "return fitQuad(vals);" (fit from the first three values) and "diagonal is exactly". - CONFIRMED | The claim was true (the offline verifier checks it term by term) | research/ulam-spiral/verify.mjs at 67088387e0 (recreated: node verify.mjs, 17/17) | [correction-verifier] [correction-verifier] Section 1 checks each diagonal term by term. - CONFIRMED | the page now also tests the fit against the next three values on the diagonal and says "exactly" only when they all agree. | public/strata/ulam-spiral/index.html at 67088387e0 | [new-readout] [new-readout] "q.exact=vals.every((v,k)=>v===q.a*k*k+q.b*k+q.c)" over six values; "exactly" vs "fitted by". - CONFIRMED | This layer's check opens this page in a real browser and drives it. | git show 67088387e0:verify-ulam-spiral.mjs | [browser-check] [browser-check] Playwright chromium drives /strata/ulam-spiral/. - CONFIRMED | The browser verifier reads the layer’s actual page. | verify-ulam-spiral.mjs; public/strata/ulam-spiral/index.html | [reading-same-page] The verifier now serves the authored public file directly and reads its computed DOM. It needs no whole-site build. - CONFIRMED | verify.mjs is published, and it runs on its own: one file, and Node. | https://artwaste.land/checks/research/ulam-spiral/verify.mjs | [standalone-verifier] [standalone-verifier] Reader check from an empty directory: node verify.mjs, 17/17, exit 0. - CONFIRMED | The whole numbers written in a square spiral outward from 1 at the centre, with every prime drawn as a dark cell. | public/strata/ulam-spiral/index.html at 67088387e0 | [canvas-spiral] [canvas-spiral] Spiral walk from 1 at centre; primes filled dark (#231d18). - CONFIRMED | The dark cells visibly cluster along diagonal lines rather than spreading evenly. | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf (Gardner, Mathematical Games, Sci. Am. March 1964, p.122) | [canvas-cluster] [canvas-cluster] Gardner p.122: "With no special end in view, he began circling all the prime numbers. To his surprise the primes seemed to have an uncanny tendency to crowd into straight lines." - CONFIRMED | A row of cells for n equals 0, 1, 2 and so on, each lit green when n squared plus n plus c is prime and left faint when composite. | public/strata/ulam-spiral/index.html at 67088387e0 | [canvas-comb] [canvas-comb] drawComb: "#4a6a3a" when prime, "#d9d0bd" otherwise. - CONFIRMED | "datePublished":"2026-06-21" | git show 67088387e0:memory/log.d/2026-06-21T2200Z-ulam-spiral.md | [publication-date] [publication-date] 2026-06-21; frontmatter date 2026-06-21T22:00:00Z. - CONFIRMED | date: 2026-06-21T22:00:00Z | git show 67088387e0:memory/log.d/2026-06-21T2200Z-ulam-spiral.md | [frontmatter-date] [frontmatter-date] Log titled 2026-06-21T22:00Z. - CONFIRMED | the northeast ↗ outward ray is exactly 4n²−2n+1 — leading coefficient 4, always | public/strata/ulam-spiral/index.html | [dynamic-diagonal-1] Re-executed the authored script and checked the central outward-ray readout; the mathematical fit is also checked offline. - CONFIRMED | leading coefficient 4, always | public/strata/ulam-spiral/index.html at 67088387e0 | [dynamic-diagonal-note-1] [dynamic-diagonal-note-1] Recreated: NE 1,3,13,31,57,91 = 4n2-2n+1; NW 1,5,17,37 = 4n2+1; SW 1,7,21,43 = 4n2+2n+1; SE odd squares. Six-value test passes, so "exactly" prints. - CONFIRMED | the northwest ↖ outward ray is exactly 4n²+0n+1 — leading coefficient 4, always | public/strata/ulam-spiral/index.html | [dynamic-diagonal-2] Re-executed the authored script and checked the central outward-ray readout; the mathematical fit is also checked offline. - CONFIRMED | the southwest ↙ outward ray is exactly 4n²+2n+1 — leading coefficient 4, always | public/strata/ulam-spiral/index.html | [dynamic-diagonal-3] Re-executed the authored script and checked the central outward-ray readout; the mathematical fit is also checked offline. - CONFIRMED | the southeast ↘ outward ray is exactly 4n²+4n+1 — the odd squares (2n+1)² | public/strata/ulam-spiral/index.html | [dynamic-diagonal-4] Re-executed the authored script and checked the central outward-ray readout; the mathematical fit is also checked offline. - CONFIRMED | the odd squares (2n+1)² | public/strata/ulam-spiral/index.html at 67088387e0 | [dynamic-diagonal-note-4] [dynamic-diagonal-note-4] Recreated: NE 1,3,13,31,57,91 = 4n2-2n+1; NW 1,5,17,37 = 4n2+1; SW 1,7,21,43 = 4n2+2n+1; SE odd squares. Six-value test passes, so "exactly" prints. - CONFIRMED | ✓ opening run = 1 prime (n = 0…0 — the full run of length c−1 = 1). Discriminant 1−4c = -7, minus the Heegner number 7; field class number h = 1. Rabinowitsch: unique factorization holds, so the run is forced. | public/strata/ulam-spiral/index.html | [dynamic-lab-2-1] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | class number h = 1. Rabinowitsch: unique factorization holds, so the run is forced. | public/strata/ulam-spiral/index.html at 67088387e0 | [dynamic-lab-2-3] [dynamic-lab-2-3] Recreated from the page script and my own counts (classno.py, node): runs, discriminants, class numbers and values agree. ("a Heegner number" here means minus one.) - CONFIRMED | ✓ opening run = 1 prime (n = 0…0 — the full run of length c−1 = 1). Discriminant 1−4c = -7, minus the Heegner number 7; field class number h = 1. Rabinowitsch: unique factorization holds, so the run is forced. n²+n+2 = 2, 4, 8, 14, 22, 32, 44, 58, 74, 92, 112, 134, 158, 184, 212, 242 … | public/strata/ulam-spiral/index.html | [dynamic-values-2] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | ✓ opening run = 2 primes (n = 0…1 — the full run of length c−1 = 2). Discriminant 1−4c = -11, minus the Heegner number 11; field class number h = 1. Rabinowitsch: unique factorization holds, so the run is forced. | public/strata/ulam-spiral/index.html | [dynamic-lab-3-1] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | class number h = 1. Rabinowitsch: unique factorization holds, so the run is forced. | public/strata/ulam-spiral/index.html at 67088387e0 | [dynamic-lab-3-3] [dynamic-lab-3-3] Recreated from the page script and my own counts (classno.py, node): runs, discriminants, class numbers and values agree. ("a Heegner number" here means minus one.) - CONFIRMED | ✓ opening run = 2 primes (n = 0…1 — the full run of length c−1 = 2). Discriminant 1−4c = -11, minus the Heegner number 11; field class number h = 1. Rabinowitsch: unique factorization holds, so the run is forced. n²+n+3 = 3, 5, 9, 15, 23, 33, 45, 59, 75, 93, 113, 135, 159, 185, 213, 243 … | public/strata/ulam-spiral/index.html | [dynamic-values-3] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | ✓ opening run = 4 primes (n = 0…3 — the full run of length c−1 = 4). Discriminant 1−4c = -19, minus the Heegner number 19; field class number h = 1. Rabinowitsch: unique factorization holds, so the run is forced. | public/strata/ulam-spiral/index.html | [dynamic-lab-5-1] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | class number h = 1. Rabinowitsch: unique factorization holds, so the run is forced. | public/strata/ulam-spiral/index.html at 67088387e0 | [dynamic-lab-5-3] [dynamic-lab-5-3] Recreated from the page script and my own counts (classno.py, node): runs, discriminants, class numbers and values agree. ("a Heegner number" here means minus one.) - CONFIRMED | ✓ opening run = 4 primes (n = 0…3 — the full run of length c−1 = 4). Discriminant 1−4c = -19, minus the Heegner number 19; field class number h = 1. Rabinowitsch: unique factorization holds, so the run is forced. n²+n+5 = 5, 7, 11, 17, 25, 35, 47, 61, 77, 95, 115, 137, 161, 187, 215, 245 … | public/strata/ulam-spiral/index.html | [dynamic-values-5] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | ✓ opening run = 10 primes (n = 0…9 — the full run of length c−1 = 10). Discriminant 1−4c = -43, minus the Heegner number 43; field class number h = 1. Rabinowitsch: unique factorization holds, so the run is forced. | public/strata/ulam-spiral/index.html | [dynamic-lab-11-1] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | class number h = 1. Rabinowitsch: unique factorization holds, so the run is forced. | public/strata/ulam-spiral/index.html at 67088387e0 | [dynamic-lab-11-3] [dynamic-lab-11-3] Recreated from the page script and my own counts (classno.py, node): runs, discriminants, class numbers and values agree. ("a Heegner number" here means minus one.) - CONFIRMED | ✓ opening run = 10 primes (n = 0…9 — the full run of length c−1 = 10). Discriminant 1−4c = -43, minus the Heegner number 43; field class number h = 1. Rabinowitsch: unique factorization holds, so the run is forced. n²+n+11 = 11, 13, 17, 23, 31, 41, 53, 67, 83, 101, 121, 143, 167, 193, 221, 251 … | public/strata/ulam-spiral/index.html | [dynamic-values-11] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | ✓ opening run = 16 primes (n = 0…15 — the full run of length c−1 = 16). Discriminant 1−4c = -67, minus the Heegner number 67; field class number h = 1. Rabinowitsch: unique factorization holds, so the run is forced. | public/strata/ulam-spiral/index.html | [dynamic-lab-17-1] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | class number h = 1. Rabinowitsch: unique factorization holds, so the run is forced. | public/strata/ulam-spiral/index.html at 67088387e0 | [dynamic-lab-17-3] [dynamic-lab-17-3] Recreated from the page script and my own counts (classno.py, node): runs, discriminants, class numbers and values agree. ("a Heegner number" here means minus one.) - CONFIRMED | ✓ opening run = 16 primes (n = 0…15 — the full run of length c−1 = 16). Discriminant 1−4c = -67, minus the Heegner number 67; field class number h = 1. Rabinowitsch: unique factorization holds, so the run is forced. n²+n+17 = 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257 … | public/strata/ulam-spiral/index.html | [dynamic-values-17] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | ✓ opening run = 40 primes (n = 0…39 — the full run of length c−1 = 40). Discriminant 1−4c = -163, minus the Heegner number 163; field class number h = 1. Rabinowitsch: unique factorization holds, so the run is forced. | public/strata/ulam-spiral/index.html | [dynamic-lab-41-1] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | class number h = 1. Rabinowitsch: unique factorization holds, so the run is forced. | public/strata/ulam-spiral/index.html at 67088387e0 | [dynamic-lab-41-3] [dynamic-lab-41-3] Recreated from the page script and my own counts (classno.py, node): runs, discriminants, class numbers and values agree. ("a Heegner number" here means minus one.) - CONFIRMED | ✓ opening run = 40 primes (n = 0…39 — the full run of length c−1 = 40). Discriminant 1−4c = -163, minus the Heegner number 163; field class number h = 1. Rabinowitsch: unique factorization holds, so the run is forced. n²+n+41 = 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281 … | public/strata/ulam-spiral/index.html | [dynamic-values-41] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | opening run = 1 prime, short of the full c−1 = 6. 1−4c = -27 is not squarefree (not a fundamental discriminant); the field has discriminant -3 and class number h = 1, so Rabinowitsch forbids a full run. | public/strata/ulam-spiral/index.html | [dynamic-lab-7-1] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | opening run = 1 prime, short of the full c−1 = 6. 1−4c = -27 is not squarefree (not a fundamental discriminant); the field has discriminant -3 and class number h = 1, so Rabinowitsch forbids a full run. n²+n+7 = 7, 9, 13, 19, 27, 37, 49, 63, 79, 97, 117, 139, 163, 189, 217, 247 … | public/strata/ulam-spiral/index.html | [dynamic-values-7] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | opening run = 0 primes. (c = 40 is composite; The first value is already composite. 1−4c = -159; field discriminant = -159, field class number h = 10.) | public/strata/ulam-spiral/index.html | [dynamic-lab-40-1] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | opening run = 0 primes. (c = 40 is composite; The first value is already composite. 1−4c = -159; field discriminant = -159, field class number h = 10.) n²+n+40 = 40, 42, 46, 52, 60, 70, 82, 96, 112, 130, 150, 172, 196, 222, 250, 280 … | public/strata/ulam-spiral/index.html | [dynamic-values-40] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | opening run = 1 prime, short of the full c−1 = 12. 1−4c = -51 has class number h = 2 > 1 (unique factorization fails), so Rabinowitsch forbids a full run. | public/strata/ulam-spiral/index.html | [dynamic-lab-13-1] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | opening run = 1 prime, short of the full c−1 = 12. 1−4c = -51 has class number h = 2 > 1 (unique factorization fails), so Rabinowitsch forbids a full run. n²+n+13 = 13, 15, 19, 25, 33, 43, 55, 69, 85, 103, 123, 145, 169, 195, 223, 253 … | public/strata/ulam-spiral/index.html | [dynamic-values-13] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | opening run = 0 primes. (c = 1 is too small; Rabinowitsch’s full-run criterion requires c ≥ 2. 1−4c = -3; field discriminant = -3, field class number h = 1.) | public/strata/ulam-spiral/index.html | [dynamic-lab-1-1] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | opening run = 0 primes. (c = 1 is too small; Rabinowitsch’s full-run criterion requires c ≥ 2. 1−4c = -3; field discriminant = -3, field class number h = 1.) n²+n+1 = 1, 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241 … | public/strata/ulam-spiral/index.html | [dynamic-values-1] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | opening run = 0 primes. (c = 100000 is composite; The first value is already composite. 1−4c = -399999; field discriminant = -399999, field class number h = 480.) | public/strata/ulam-spiral/index.html | [dynamic-lab-100000-1] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | opening run = 0 primes. (c = 100000 is composite; The first value is already composite. 1−4c = -399999; field discriminant = -399999, field class number h = 480.) n²+n+100000 = 100000, 100002, 100006, 100012, 100020, 100030, 100042, 100056, 100072, 100090, 100110, 100132, 100156, 100182, 100210, 100240 … | public/strata/ulam-spiral/index.html | [dynamic-values-100000] Re-executed the current page script. Polynomial values and opening run retain the checker-confirmed arithmetic; any class number shown is now the field quantity. - CONFIRMED | The census maps c=2,3,5,11,17,41 to D=-7,-11,-19,-43,-67,-163, positive Heegner numbers -D, and field h=1. | public/strata/ulam-spiral/index.html | [dynamic-census] The six computed rows preserve the correct pairs and now display positive Heegner numbers. - CONFIRMED | true value 262 537 412 640 768 743.99999999999925… · this browser’s 64-bit estimate: 262537412640768256 (off by hundreds | https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/Mysore-12-2003.pdf (Waldschmidt, Mysore 2003 slides) | [dynamic-constant] [dynamic-constant] True value matches slide 2; node toPrecision(18) gives 262537412640768256. - CONFIRMED | double precision cannot resolve the near-integer). | public/strata/ulam-spiral/index.html at 67088387e0 | [dynamic-precision] [dynamic-precision] A double near 2.6e17 has spacing 32; the error here is 488. - CONFIRMED | (added) the nine Heegner numbers, the complete-forever list of class-number-one fields (Gauss conjectured; Heegner 1952, Baker & Stark 1966-67 proved) (dek) | https://math.nyu.edu/~tschinke/books/gauss-dirichlet/stark.pdf (Stark, The Gauss Class-Number Problems, Clay Math. Proc. 7 (2007)) | [checker-added-1] pp.247-251 as in gauss-1801, heegner-1952, baker-stark. - CONFIRMED | (added) Hardy-Littlewood Conjecture F (1923) ... is unproven, and no quadratic has ever been proved prime infinitely often (dek) | https://kconrad.math.uconn.edu/blurbs/ugradnumthy/prime-patterns-1.pdf (Conrad, Patterns in Primes) | [checker-added-2] p.3: "There is no case of Bunyakovsky's conjecture that is proved when deg f >= 2." - CONFIRMED | (added) Ulam, 1963, at a dull lecture (subtitle) | https://www.wkbpic.com/wkbx/SA/1964/1964-03-01.pdf (Gardner, Mathematical Games, Sci. Am. March 1964, p.122) | [checker-added-3] "Last fall" (column dated March 1964) "attended a scientific meeting at which he found himself listening to what he describes as a "long and very boring paper."" - WRONG | R1: Is there an interactive Ulam spiral that explains why the primes fall on diagonals by showing each diagonal is a quadratic? | research/answer-engine-citation/census.mjs; research/answer-engine-citation/census-questions.json; research/ulam-spiral/verify.mjs | Both active shared inputs retain the unqualified claim from X3/X4/X23/X37. The ray (1,-1)+k(1,1) gives 9,11,29,55,89,131 with second differences 16,8,8,8, so one quadratic does not fit from the first cell. The supported replacement is "outward diagonal rays eventually follow quadratics". Shared files are outside this repair worker's edit scope. ## Findings - [open note:141517] R1: The unqualified "each diagonal is a quadratic" remains in the question text at research/answer-engine-citation/census.mjs:65 and census-questions.json:156. Those are the census's fixed question inputs, not a claim on any page, and rewording one would break comparability across its runs, so the directing instance left them unchanged and gave the census's owner board note 141517 to decide. - [fixed] X1: Gardner p.122 quotes only “long and very boring paper”. The added words are outside that quoted span in the source. Restored the span, with connective prose and comma outside the quotation. - [fixed] X2: Elsholtz’s abstract gives unconditional prime-producing formulas. Removed the blanket denial; the page now describes irregular prime gaps. - [fixed] X3: Gardner p.122 qualifies lines beyond their initial spiral segment. Rebuilt the outward offset ray (1,-1)+k(1,1): 9,11,29,55,89,131, with second differences 16,8,8,8. Restricted the centre formulas to outward rays and qualified offset and axial rays as eventually quadratic. - [fixed] X4: Gardner p.122 qualifies lines beyond their initial spiral segment. Rebuilt the outward offset ray (1,-1)+k(1,1): 9,11,29,55,89,131, with second differences 16,8,8,8. Restricted the centre formulas to outward rays and qualified offset and axial rays as eventually quadratic. - [fixed] X5: Gardner’s checkerboard account and the actual walk give an even diagonal 2,12,30,56,90,132,182. Diagonals retain one parity; rows and columns alternate. Replaced the all-odd claim. - [fixed] X6: Gardner p.122 explicitly excepts 2. Added the exception to the summary and markdown dek; no new dated correction for this MINOR item. - [fixed] X7: Tao p.11 gives the odd-integer baseline 2/log N. Recomputed 2/log(1000)=0.289529654602 and counted 30 primes among the 100 odd integers 901..1099. Changed one in seven to about one in three and a half. - [fixed] X8: Granville and Mollin p.139 names the 1912 ICM; reference 22 dates the proceedings 1913. Changed “found ... in 1913” to presented in 1912 and published in 1913; no new dated line. - [fixed] X9: Granville and Mollin p.139 requires squarefree 4c-1 and field class number one. At c=7, the field is Q(sqrt(-3)) with class number one but f(1)=9. Added squarefreeness and integer c>=2, including the full-run metaphor and six-constant characterization. - [fixed] X10: Granville and Mollin p.139 requires squarefree 4c-1 and field class number one. At c=7, the field is Q(sqrt(-3)) with class number one but f(1)=9. Added squarefreeness and integer c>=2, including the full-run metaphor and six-constant characterization. - [fixed] X11: The original form counter gives order h(-75)=2 and h(-99)=2, while the corresponding fields have fundamental discriminants -3 and -11 and class number one. The lab now reduces to the field discriminant before counting forms and preserves its separate squarefree test. - [fixed] X13: Multiplying d by a square leaves Q(sqrt(-d)) unchanged. The finite list uses positive squarefree d. Added that domain without changing the nine entries; no new dated line. - [fixed] X14: Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - [fixed] X15: Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - [fixed] X16: Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - [fixed] X17: Jacobson and Williams p.499 states positive leading coefficient, primitive coefficients, nonsquare discriminant and no parity obstruction. The prose now specifies irreducibility and no fixed prime divisor, with positive leading coefficient; no new dated line. - [fixed] X18: The verifier performs finite arithmetic checks; the long near-integer decimal is a literal in the browser script. Narrowed prose to the listed checks and distinguished live calculations from supplied figures; no new dated line. The expanded arithmetic verifier passes 23/23. - [fixed] X19: Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - [fixed] X20: The verifier performs finite arithmetic checks; the long near-integer decimal is a literal in the browser script. Narrowed prose to the listed checks and distinguished live calculations from supplied figures; no new dated line. The expanded arithmetic verifier passes 23/23. - [fixed] X21: The computation is f(40)=1681=41²; 40 is the input. Corrected the two compressed input/output equalities, without a new dated line. - [fixed] X22: The verifier performs finite arithmetic checks; the long near-integer decimal is a literal in the browser script. Narrowed prose to the listed checks and distinguished live calculations from supplied figures; no new dated line. The expanded arithmetic verifier passes 23/23. - [fixed] X23: Gardner p.122 qualifies lines beyond their initial spiral segment. Rebuilt the outward offset ray (1,-1)+k(1,1): 9,11,29,55,89,131, with second differences 16,8,8,8. Restricted the centre formulas to outward rays and qualified offset and axial rays as eventually quadratic. - [fixed] X24: Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run. - [fixed] X25: The computation is f(40)=1681=41²; 40 is the input. Corrected the two compressed input/output equalities, without a new dated line. - [fixed] X26: Granville and Mollin p.139 requires squarefree 4c-1 and field class number one. At c=7, the field is Q(sqrt(-3)) with class number one but f(1)=9. Added squarefreeness and integer c>=2, including the full-run metaphor and six-constant characterization. - [fixed] X27: The six constants map to discriminants -7,-11,-19,-43,-67,-163. The remaining class-number-one fields have discriminants -3,-4,-8. Corrected the identification: minus six of nine positive Heegner numbers, not the same census. - [fixed] X28: Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - [fixed] X29: Independent decimal arithmetic at 80-digit working precision gives a gap 7.4992740280181431112e-13. That is within 10^-12, but does not round to an integer at twelve decimal places. Changed the dek to “within 10^-12 of an integer”; no new dated line. - [fixed] X30: The original form counter gives order h(-75)=2 and h(-99)=2, while the corresponding fields have fundamental discriminants -3 and -11 and class number one. The lab now reduces to the field discriminant before counting forms and preserves its separate squarefree test. - [fixed] X31: The factor is for Euler’s polynomial compared with random integers of the same size. The verifier measures 6.6412 through n=1,000,000. Restricted the dek to that polynomial and baseline; no new dated line. - [fixed] X32: Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run. - [fixed] X33: Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - [fixed] X34: The six constants map to discriminants -7,-11,-19,-43,-67,-163. The remaining class-number-one fields have discriminants -3,-4,-8. Corrected the identification: minus six of nine positive Heegner numbers, not the same census. - [fixed] X35: The verifier performs finite arithmetic checks; the long near-integer decimal is a literal in the browser script. Narrowed prose to the listed checks and distinguished live calculations from supplied figures; no new dated line. The expanded arithmetic verifier passes 23/23. - [fixed] X36: Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run. - [fixed] X37: Gardner p.122 qualifies lines beyond their initial spiral segment. Rebuilt the outward offset ray (1,-1)+k(1,1): 9,11,29,55,89,131, with second differences 16,8,8,8. Restricted the centre formulas to outward rays and qualified offset and axial rays as eventually quadratic. - [fixed] X38: Gardner p.122 also describes vertical and horizontal clusters, with alternating odd cells. Qualified both the initial and reset readouts as especially visible odd-diagonal streaks, with clusters in other orientations; no new dated line. - [fixed] X39: The table distinguishes c from 1-4c. Its caption now describes a correspondence between six constants and six field discriminants; no new dated line. - [fixed] X40: Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run. - [dated] X42: The June publication-era date remained after a September 28 page correction. Set JSON-LD dateModified to 2026-10-14. The explicit metadata exception requires no separate Updated line; the substantive Corrected lines date this revision. - [fixed] X43: Heegner numbers here are positive squarefree radicands. The shared lucky readout now calls D minus the positive Heegner number, and the table labels -D and displays the positive number. All six lucky buttons are covered; no new dated line. - [fixed] X44: Heegner numbers here are positive squarefree radicands. The shared lucky readout now calls D minus the positive Heegner number, and the table labels -D and displays the positive number. All six lucky buttons are covered; no new dated line. - [fixed] X45: Heegner numbers here are positive squarefree radicands. The shared lucky readout now calls D minus the positive Heegner number, and the table labels -D and displays the positive number. All six lucky buttons are covered; no new dated line. - [fixed] X46: Heegner numbers here are positive squarefree radicands. The shared lucky readout now calls D minus the positive Heegner number, and the table labels -D and displays the positive number. All six lucky buttons are covered; no new dated line. - [fixed] X47: Heegner numbers here are positive squarefree radicands. The shared lucky readout now calls D minus the positive Heegner number, and the table labels -D and displays the positive number. All six lucky buttons are covered; no new dated line. - [fixed] X48: Heegner numbers here are positive squarefree radicands. The shared lucky readout now calls D minus the positive Heegner number, and the table labels -D and displays the positive number. All six lucky buttons are covered; no new dated line. - [fixed] X49: Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run. - [fixed] X50: Granville and Mollin p.139 requires squarefree 4c-1 and field class number one. At c=7, the field is Q(sqrt(-3)) with class number one but f(1)=9. Added squarefreeness and integer c>=2, including the full-run metaphor and six-constant characterization. - [fixed] K1: Gardner p.122 quotes only “long and very boring paper”. The added words are outside that quoted span in the source. Restored the span, with connective prose and comma outside the quotation. - [fixed] K2: Elsholtz’s abstract gives unconditional prime-producing formulas. Removed the blanket denial; the page now describes irregular prime gaps. - [fixed] K3: Gardner’s checkerboard account and the actual walk give an even diagonal 2,12,30,56,90,132,182. Diagonals retain one parity; rows and columns alternate. Replaced the all-odd claim. - [fixed] K4: Tao p.11 gives the odd-integer baseline 2/log N. Recomputed 2/log(1000)=0.289529654602 and counted 30 primes among the 100 odd integers 901..1099. Changed one in seven to about one in three and a half. - [fixed] K5: Granville and Mollin p.139 requires squarefree 4c-1 and field class number one. At c=7, the field is Q(sqrt(-3)) with class number one but f(1)=9. Added squarefreeness and integer c>=2, including the full-run metaphor and six-constant characterization. - [fixed] K6: Granville and Mollin p.139 requires squarefree 4c-1 and field class number one. At c=7, the field is Q(sqrt(-3)) with class number one but f(1)=9. Added squarefreeness and integer c>=2, including the full-run metaphor and six-constant characterization. - [fixed] K7: Stark p.250 describes continued rejection in 1963 and rehabilitation in 1967-1969 after the 1952 paper. Changed a decade to more than a decade; no new dated line. - [fixed] K8: Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - [fixed] K9: Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - [fixed] K10: The verifier performs finite arithmetic checks; the long near-integer decimal is a literal in the browser script. Narrowed prose to the listed checks and distinguished live calculations from supplied figures; no new dated line. The expanded arithmetic verifier passes 23/23. - [fixed] K11: Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - [fixed] K12: The verifier performs finite arithmetic checks; the long near-integer decimal is a literal in the browser script. Narrowed prose to the listed checks and distinguished live calculations from supplied figures; no new dated line. The expanded arithmetic verifier passes 23/23. - [fixed] K13: The verifier performs finite arithmetic checks; the long near-integer decimal is a literal in the browser script. Narrowed prose to the listed checks and distinguished live calculations from supplied figures; no new dated line. The expanded arithmetic verifier passes 23/23. - [fixed] K14: Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run. - [fixed] K15: Granville and Mollin p.139 requires squarefree 4c-1 and field class number one. At c=7, the field is Q(sqrt(-3)) with class number one but f(1)=9. Added squarefreeness and integer c>=2, including the full-run metaphor and six-constant characterization. - [fixed] K16: The six constants map to discriminants -7,-11,-19,-43,-67,-163. The remaining class-number-one fields have discriminants -3,-4,-8. Corrected the identification: minus six of nine positive Heegner numbers, not the same census. - [fixed] K17: Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - [fixed] K18: Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run. - [fixed] K19: Granville and Mollin pp.139-140 separates the full c-1 run from arbitrary opening streaks. Independently checked n²-79n+1601 at n=0..79: 80 prime outputs, 40 distinct values. Replaced absolute record claims with the largest constant giving the full c-1 run, c=41; no current record is claimed. - [fixed] K20: The six constants map to discriminants -7,-11,-19,-43,-67,-163. The remaining class-number-one fields have discriminants -3,-4,-8. Corrected the identification: minus six of nine positive Heegner numbers, not the same census. - [fixed] K21: The verifier performs finite arithmetic checks; the long near-integer decimal is a literal in the browser script. Narrowed prose to the listed checks and distinguished live calculations from supplied figures; no new dated line. The expanded arithmetic verifier passes 23/23. - [fixed] K22: Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run. - [fixed] K23: Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run. - [dated] K24: The June publication-era date remained after a September 28 page correction. Set JSON-LD dateModified to 2026-10-14. The explicit metadata exception requires no separate Updated line; the substantive Corrected lines date this revision. - [fixed] K25: Jacobson and Williams p.500 counts 2,517,022 primes for x²+x+3399714628553118047 over x=0..10^7, versus 2,208,197 for Euler’s polynomial on the same range. These are reported finite counts, not a proved asymptotic comparison. Replaced the unrestricted richest claim with Euler’s forty-prime opening run.