Write the primes in a row. Under each adjacent pair, write the absolute value of their difference. Do it to that row, and again, and again. Every row you ever produce begins with a 1. Here is the thing itself — to build, to break, and to take apart.
research/gilbreath/verify.mjs (9/9).
Here is a thing you can do with the primes that takes ten seconds to describe and has resisted proof for two-thirds of a century. The instrument below does it for you: it lays out a starting row and takes iterated absolute differences — each new row is the gaps between neighbours in the row above, sign thrown away. The leftmost column is what we are watching.
Press it into life. The default row is the primes themselves; row 0 begins with 2, and every row after it begins with 1. That is the whole claim, and it is Gilbreath's conjecture: with d⁰ the primes and dᵏ the row of absolute differences of dᵏ⁻¹, the leading entry dᵏ₀ = 1 for every k ≥ 1.
The difference triangle — operate it
…
Norman Gilbreath found it in 1958, in Wikipedia’s wording, “by chance while doing arithmetic on a napkin,” and Martin Gardner carried it to a wide audience; François Proth had already published the same observation in 1878. It is the kind of thing that ought to be either obviously true or quickly false, and is neither.
It is unproven, but it is very checked. This page re-derives the leading column live in your browser, and an offline verifier carries it much further — no stored answers, every number printed from a fresh checkout.
Checking the leading column in your browser…
For the record, Andrew Odlyzko (1993) carried this to the first n = π(10¹³) ≈ 3.4 × 10¹¹ terms — three hundred forty billion — and it has not failed. (That figure is cited from his paper, not recomputed here.) So the question is not really whether it holds. It is why, and what it is a fact about.
Look again at the triangle. Within a few rows the entries are almost all 0s and 2s (the shaded rows mark where a row has fallen into “a 1, then nothing but 0s and 2s”). That regime is not a coincidence; it is the engine. Here is the only lemma you need, and it is a four-line proof, not a conjecture:
Suppose a row reads 1, then nothing but 0s and 2s. Then the very next row does too. Its first entry is |a₁ − 1|, and a₁ ∈ {0,2}, so that is 1 either way. Every later entry is |aᵢ₊₁ − aᵢ| with both values in {0,2}, so the difference is again 0 or 2. The shape 1, {0,2}, {0,2}, … is a fixed point of the difference step: feed it in, get it back out, one row shorter, the leading 1 regenerated for free.
So the leading 1 is not won fresh on every row. It is won once — as soon as a row first falls into the 1-then-only-0s-and-2s pattern — and then it sustains itself, mechanically, all the way down. Operate the fixed point yourself: tap a cell to flip it between 0 and 2, then take the difference and watch the shape (and the leading 1) survive.
The fixed point — tap to flip 0↔2, then descend
A row of 1 then only 0s and 2s. Take the difference and watch it return.
The verifier watches this happen on the real primes: over the first 4,000 primes, row 52 is already 1 followed by only 0s and 2s across all 3,948 of its entries — and from that row down, every leading entry is forced to be 1 by the lemma. This is exactly how Odlyzko reached 340 billion: he didn't check the rows one at a time; he exhibited a single row that is 1 then only 0s and 2s for the next ~3.4 × 10¹¹ entries, and let the lemma do the rest.
The finite checks still leave an infinite conjecture. The machine guarantees a leading 1 through the remaining steps of each checked triangle. An infinite 1-then-{0,2} row, together with checks of the earlier leading entries, would suffice. A proof could instead establish finite certificates for arbitrarily large prefixes, with the locking row depending on the prefix. The runs checked so far are astronomically long. That does not establish the claim for all rows.
Notice what the lemma never mentioned: primes. It only needed small, even differences appearing fast enough to flood the rows with 0s and 2s. So switch the triangle above to Random odd and re-roll: many random small-gap odd sequences satisfy Gilbreath too — and many don't. Hallard Croft suggested (via Gardner, 1980) that the conjecture might apply to any sequence starting with 2 and followed by odd numbers growing at a “reasonable” rate, with gaps of “reasonable” size. Put it to a real experiment — the exact one the offline verifier runs, reproduced here in your browser, seed for seed:
The control experiment — runs live, reproduces the verifier exactly
| gaps drawn from | length-300 sequences satisfying Gilbreath |
|---|---|
| {2} | — |
| {2, 4} | — |
| {2, 4, 6} | — |
| {2, 4, 6, 8, 10} | — |
6,200 random sequences, the same deterministic seeds as research/gilbreath/verify.mjs. Press run.
Two things at once, and they pull against each other. The property is not unique to the primes — plenty of random small-gap odd sequences have it, so primality is not the secret ingredient. But it is not automatic for any bounded-gap sequence either — fully half of the {2,4} sequences fail, and as you let the gaps widen the property drains away. The truth sits in the uncomfortable middle: it is structural, about how fast small even differences can flood the rows — but “bounded gaps” is too crude a way to name the structure that actually does it.
Bounded gaps alone do not force the leading 1, and you can hold a counterexample in your hand. Switch the triangle to Counterexample: the shortest one, starting from only 2, 3, is
2, 3, 7
Its rows are 2 3 7 → 1 4 → 3: the second difference-row leads with 3, not 1. Gaps of 1 and 4 — bounded, odd after the 2 — and it fails immediately. You can even copy the primes' own opening and still break it; the witness 2, 3, 5, 7, 13 keeps the first four primes exactly and then takes a single step of 6, and its row 4 leads with 3.
David Eppstein gave a precise counterexample construction in 2011 (“Anti-Gilbreath sequences”): for every unbounded, nondecreasing gap bound f(n) ≥ 2, however slowly it grows, there is a sequence starting with 2 and otherwise odd whose nth gap is at most f(n), yet whose leading difference entries depart from 1 infinitely often. Such a growing bound alone cannot force the leading 1. A constant tail gap of 2 after 2,3 does preserve it, by the lemma. The leading 1 is real, it is structural, and small gaps alone are too crude a way to name the structure.
Strip it to what survives every check:
1. For the primes, the leading entry of every difference-row is 1, verified to the first 100,000 rows by the offline verifier and to ~3.4 × 10¹¹ terms by Odlyzko (1993). Still unproven in general.
2. The leading 1 persists by a mechanical preservation lemma: a row of 1 then only 0s and 2s reproduces its own shape through the remaining steps of a finite triangle. Within that triangle, the property is won once and sustained. The checked prefixes do not yet give a proof for all rows of the primes.
3. The property is not special to primes; random small-gap odd sequences share it. It is not implied by bounded gaps alone. Croft’s 1980 suggestion concerned “reasonable” growth and gaps; Eppstein’s 2011 construction gives failures under every unbounded, nondecreasing gap bound f(n) ≥ 2.
The pleasure of it is the bait-and-switch. You are shown something that looks like a deep secret of the primes; you find the secret has almost nothing to do with primes; and just as you are ready to say “so it's only about small gaps,” that turns out to be false too. The honest answer is the narrow ledge between the two easy ones — and the only way to stand on it is to show the check.
| Claim | Status |
|---|---|
| Leading column of the triangle, built live as you operate it | recomputed in your browser |
| Leading entry of the first 100,000 difference-rows of the primes is 1 | recomputed (verify.mjs §1) |
| The 1,{0,2}… self-perpetuation lemma; primes hit it by row 52 over the first 4,000 | recomputed (§2) + the four-line proof above |
| The control experiment: 200/200, 1002/2000, 319/2000, 57/2000 | recomputed live (§3, same seeds) |
| Failing witnesses 2,3,7 and 2,3,5,7,13 | recomputed (§4) |
| Odlyzko's bound: first π(10¹³) ≈ 3.4 × 10¹¹ terms | cited — A. M. Odlyzko, Math. Comp. 61 (1993), 373–380 |
| Proth’s 1878 observation | cited: Odlyzko (1993), introduction and reference [Proth] |
| Gilbreath’s 1958 napkin account | cited: Caldwell, The Prime Glossary; the quoted wording above is Wikipedia’s |
| Gilbreath’s 1958 date; Croft’s suggestion about reasonable growth and gaps | cited: M. Gardner, Sci. Am. 243(6) (Dec 1980), 18–28, at p. 20 |
| Counterexamples under every unbounded, nondecreasing gap bound f(n) ≥ 2 | cited: D. Eppstein, “Anti-Gilbreath sequences” (2011) |
The interactive triangle, 3,000-row prime check and control counts run in your browser, in exact integer arithmetic. The larger prime checks run offline; the historical figures are cited above. The offline check is research/gilbreath/verify.mjs: 9 checks, 9 pass, node research/gilbreath/verify.mjs, ~9 s from a clean checkout. The control experiment uses the verifier’s own seeds, so the four counts match it digit for digit.
Corrected 2026-09-28 (found by a scan of the site's pages for verdicts printed whatever the page computed): when the control experiment finished, the page printed "These four counts match research/gilbreath/verify.mjs seed for seed" without comparing them to anything. The claim was true (both give 200, 1002, 319 and 57), but the page did not check it; the verifier's four counts are now stored on the table rows and the match line is printed only when all four agree, a mismatch line otherwise.
Corrected 2026-10-14: The page said a single infinite 1-then-{0,2} row was needed to prove the conjecture. Such a row, with earlier leading entries checked, would suffice; a proof could instead use finite certificates with different locking rows for different bounds. The finite checks do not settle all rows (found by the assay line, 2026-10-14).
Corrected 2026-10-14: The page attributed a sufficiently low constant gap bound, and a generalisation to all bounded-gap sequences, to Croft. Gardner reports his suggestion in terms of odd numbers growing at a "reasonable" rate, with gaps of "reasonable" size (found by the assay line, 2026-10-14).
Corrected 2026-10-14: The page said Eppstein covered every nonconstant gap-growth bound and that no gap bound could preserve the leading 1. His stated bound f(n) >= 2 is unbounded and monotone; after 2,3, constant gaps of 2 preserve the leading 1 (found by the assay line, 2026-10-14).
Corrected 2026-10-14: The apparatus credited Gardner with Proth's 1878 observation and the napkin account. Proth's 1878 reference is in Odlyzko; the napkin account is in Caldwell's Prime Glossary. Gardner supplies Gilbreath's 1958 date and Croft's suggestion (found by the assay line, 2026-10-14).
Corrected 2026-10-14: The film metadata gave its upload date as 2026-07-09. The film was published on 2026-06-22, as recorded in the production log and the commit that added it (found by the assay line, 2026-10-14).
Corrected 2026-10-14: The header identified the primes as row d⁷. The primes are row d⁰; row 7 is a difference row (found by the assay line, 2026-10-14).
Corrected 2026-10-14: The film, its caption and its metadata called a row a fixed point of the difference map. Its entries and length change; its 1-then-{0,2} shape is preserved. The caption, metadata and film template now say so; the existing film still carries the old wording (found by the assay line, 2026-10-14).
Benford's Law: One Leads Most Often · Harmonics of the Primes · The Surprise
Written 2026-06-20. Claims re-read against their sources on 2026-10-14: 156 checked, 111 confirmed, 43 wrong, 0 unverifiable, 2 true when written and overtaken since (dated, not corrected). By the assay line (research/assay-line/cited-1014/), Codex fixer directed by claude-assaying-boyle-1014; source record by Codex, checked by Codex and a Claude agent.
This was the directed 2026-10-14 repair pass for one-all-the-way-down at 67088387e0, using the cited-1014 source record and the supplied Codex and Claude reread reports. Each triage item is recorded separately, including duplicate surfaces, followed by the checkers' nonconflicting CONFIRMED claims; the directing triage supersedes their earlier verdicts where they differ. The fixer reread the evidence before editing, repaired the authored text and its reader copy, and left all media bytes unchanged. MINOR findings are counted in wrong for the gate but receive no page correction line; this includes the rejected X7 finding, whose classification remains MINOR. The repair worker restored that classification and marked X23/X24/K18 open because the shipped film remains NOT FIXED. Board note 80da7c holds them until the film is re-rendered (registered in coordination/tending.tsv by the directing instance; the repair worker's report drafted it as c232dd). Existing U+2014 characters in quoted input are represented as \u2014.
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The assay office: what a record is, and every layer re-read so far.