A finite object with an infinite consequence
Twenty-Nine Numbers Against Infinity
A positive-definite integer quadratic form represents every positive integer if it represents one exact list of 29. Test forms yourself, then climb the escalation branch whose successive missing values end at 290.
The deciding object, in full
A form takes an integer vector and returns one integer. The question was how many outputs must be checked before every positive integer is forced. Not the first 290. These 29.
You can hand a form to the instrument below. It does not sample. For every value at most 290, it searches the entire positive-definite ellipsoid and keeps an explicit vector. One caveat, stated here rather than buried: a search that would run past six million nodes is aborted, and an aborted search asserts nothing at all, neither presence nor absence.
Put a form against the list
Start with four squares and press Test. Then choose the form marked “misses 7”. One cell is enough to stop the certificate.
Build the last obstruction
The list was not guessed all at once. Add one new vector whose squared length is the first missing value, then ask for the next missing value. This branch reaches 290.
first missing: 1
One lattice, two bases
a = x + z, b = y + z, c = z
The manuscript prints a² + 2b² + 4c² - ac - bc. The determinant-one substitution above turns it into the displayed B. No represented value changes.
Let 290 in
Add 290w². The browser checks the 29 targets again. The theorem, not the scan by itself, supplies every positive integer.
Isolate 290
Instead add four 291-squares and one 581-square. Enter any positive integer at most 999999.
The check
Current form, recomputed live
Waiting for the first search.
Fixed source anchors
Bhargava-Hanke preprint: the exact 29-value list, the truant-290 lattice, and the isolate construction.
Bhargava 2000: the nine-value integer-matrix list and the form [1,1,1,9] missing 7.
The 29 and nine values, 291, and 581 are published exact integers displayed here, not numbers derived by the browser.
What the search certifies
The doubled Gram matrix is Hii=2cii, Hij=cij. Exact integer leading minors test positive definiteness. Exact rational LDL bounds then visit every vector with Q(x) ≤ 290. A red cell is shown only after that search completes.
The universality verdict invokes the published theorem. This page does not reproduce its full escalator classification or modular-form computation.
Uncertainty and free choices
- No displayed arithmetic has measurement uncertainty. It is exact.
- The entered coefficients, rank, preset, and test integer are reader choices. The escalation branch is not: it is fixed to the one in the manuscript, and a different admissible branch would give different intermediate forms.
- The six-million-node abort cap is a free choice made here, not a fact about the mathematics. A search that hits it returns no verdict, and the page says so rather than reporting an absence.
- The zero-click default, four squares, comes back green on all 29. That green is real but it is not evidence for the 290 theorem: four squares is universal by Lagrange, independently of anything on this page.
- The basis for B is a presentation choice. Its determinant-one change is shown.
- Witnesses are the first found in a fixed coordinate order. They are not unique.
- The Bhargava-Hanke PDF has no date or version field. Current 2026 literature still cites it as a preprint.
- Four squares makes the first preset universal independently, so its green result demonstrates the instrument, not the sharpness of 29.
Why the 291 and 581 ending covers every integer except 290
Write n = 291q + r, with 0 ≤ r ≤ 290. If r < 290, the five-variable base form represents r, and Lagrange's four-square theorem writes q as four squares. If r = 290 and n ≠ 290, then q ≥ 1 and n = 291(q - 1) + 581. At n = 290, every new square must be zero, leaving the base form, which misses 290.