Dittert’s permanent inequality
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For a nonnegative n×n matrix of total entry sum n, the Dittert functional is the product of row sums plus the product of column sums minus the permanent. Is it at most 2−n!/nⁿ, with equality only at the constant matrix? The recorded target is n=16.
Why it matters and what counts as progress
Why it matters: A new dimension of the Dittert inequality tests scaling methods for the permanent and the uniqueness of the constant maximizer.
What counts as progress: Prove the bound and unique maximizer in a new dimension or develop a scaling argument covering further dimensions.
How the problem developed
- 1987 · Partial result [1]
Paul H. H. H. Hwang Proved Dittert’s conjecture in dimension three and derived necessary conditions for equality in higher dimensions.
- 2012 · Partial result [2]
Gheorghe A. Cheon and Ian M. Wanless Established further structural restrictions on possible counterexamples; the conjecture remained open in dimensions at least four.
- 21 Jul 2026 · Reported claim [3]
Arman Kafidov, with GPT-5.6 The preprint proves the conjecture in dimension 16 and reports that a separate recent preprint handles dimensions at least 17.
AI contributions 2
AI-generated certificate candidate for Dittert’s dimension-five inequality
A released AI-generated sum-of-squares and Lean candidate encodes Φ(A)≤1226/625−||A−U₅||²_F/625 for nonnegative 5×5 matrices whose entries sum to five. If its encoding matches the original problem, it implies unique equality at U₅.
Dittert's conjecture in dimension 16 via a joint-deficit scaling lemma
Kafidov claims that, for every nonnegative 16-by-16 matrix with total entry sum 16, the Dittert functional (product of row sums plus product of column sums minus permanent) is at most 2-16!/16^16, with equality only at the constant matrix. The new joint-deficit scaling lemma makes a near maximizer doubly superstochastic after a sharper dilation and excludes a boundary maximizer at dimension 16.
Sources
- Suk Geun Hwang, On a conjecture of E. Dittert
- Gi-Sang Cheon and Ian M. Wanless, Some results towards the Dittert conjecture on permanents
- Kafidov: Dittert's conjecture in dimension 16
- Borges, Dittert dimension-five candidate
- Earlier independent dimension-five proof status
- Earlier dimension-five exact certificate
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AI-generated certificate candidate for Dittert’s dimension-five inequality