The ladder of largest primitive determinants of 3 × 3 matrices over {0,1,…,k}

Abstract

For A ∈ ℤ^(3 × 3) let g(A) be the greatest common divisor of the nine 2 × 2 minors of A; call A primitive when g(A)=1, and let Sₖ be the set of absolute determinants of primitive matrices with entries in {0,1,…,k}. Finch, in the course of determining how close to singular a unimodular 4 × 4 matrix can be, was led to Sₖ; he reported that its five largest elements appear to be k(2k²-2k+1), 2k²(k-1), k³+(k-1)³, k(2k²-3k+2), (2k²-k+1)(k-1), confirmed that nothing lies between them up through k=7, and invited an extension of the search to k=8,9,…. We carry that extension out for 8 ≤ k ≤ 20. We then continue the ladder by three further levels, k(2k-1)(k-1), k²(2k-3) and k³+(k-1)³-(k-1)², exact for 5 ≤ k ≤ 20, and continue the descent below them by four more. The last of these is not a single polynomial: writing N=N(k)=2k(k²-2k+2), we show that NinSₖ for every odd k ≥ 3 while NnotinSₖ for the even k of the swept range 6 ≤ k ≤ 20, so that there the ninth largest primitive determinant is N for odd k and N-1 for even k. The obstruction is exhibited: a matrix of determinant N carries the minors k² and (k-1)(k-2), whose gcd is 1 exactly when k is odd. Every listed value is attained by an explicit matrix template valid for all k, not merely for the range searched. The enabling tool is a proved pruning bound that replaces the (k+1)⁹ box by a sweep over its (k+1)⁶ row pairs, of which exactly 504 survive at every k in the measured range. Every theorem below is machine-checked in Lean 4; the few measurements quoted outside the formal development are labelled as such.

Open review

This founding-collection manuscript received AI review before publication. Independent human review is open. Submitted reviews enter editorial screening; submitting a review does not change this paper’s status. Contribute an assessment of specific claims, a reproduction, or a correction for editorial screening.

Archived files

1. Version 1 · current (opens in a new tab)

   Source snapshot 2026-08-30 15:34 UTC

   File fingerprint

   4b6c0717b0243e351c4bf2324504b6c16f029bb05bbca14d63410b8b36663266