Foregger’s powers conjecture

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Minc’s list · Conjecture 17

Foregger’s powers conjecture

Proved by others Candidate proof posted in September 2026

Posed by
Foregger
First posed
1978

Statement

Posed by T. H. Foregger, who communicated it to Minc; it is Conjecture 17 in Minc’s 1978 book.

Conjecture 17. Let \(\Omega_n\) be the set of \(n\times n\) doubly stochastic matrices. For every positive integer \(n\) there is an integer \(k=k(n)\) such that

\[ \operatorname{per}(A^k)\le\operatorname{per}A\qquad\text{for all }A\in\Omega_n. \]

As printed, the statement is trivially true with \(k=1\); the intended conjecture asks for a single exponent \(k(n)>1\) that works for every \(A\in\Omega_n\).

Resolution

The conjecture was proved in an AI-assisted proof posted on GitHub by infinityscroll on 7 September 2026. It establishes more: for every \(n\) there is an integer \(K(n)\ge2\) such that

\[ \operatorname{per}(A^k)\le\operatorname{per}A\qquad\text{for all }A\in\Omega_n\text{ and all }k\ge K(n). \]

No explicit value of \(K(n)\) is given. The author presents the work as a candidate proof awaiting external validation. We audited the argument independently and found no gap.

We had independently proved the same theorem by a different route, but this proof was made public first. Our paper has not been posted.

Read their proof ↗

Earlier progress

  • Chang (1984) proved \(\operatorname{per}(S^{2^k})\le\operatorname{per}S\) for all large \(k\) whenever every entry of \(S\in\Omega_n\) is at least a fixed \(c>0\).
  • Chang (1990) proved the case \(n=3\), and showed that \(\operatorname{per}A\ge\tfrac12\) implies \(\operatorname{per}(A^m)\le\operatorname{per}A\) for every \(m\ge2\).
  • Melnykova (2012) proved that each individual \(A\) satisfies the inequality for all large \(k\), with a bound depending on the smallest nonzero entry of \(A\).
  • The exponent \(k=2\) fails for every \(n\ge4\), by an example with \(\operatorname{per}A=\tfrac18<\tfrac9{64}=\operatorname{per}(A^2)\) given by Van Name in 2022.

References

  1. infinityscroll, A uniform eventual permanent-power inequality, research draft (2026). github.com/infinityscroll/foregger-1978-candidate-proof
  2. H. Minc, Permanents, Encyclopedia of Mathematics and its Applications 6, Addison-Wesley, 1978.
  3. D. K. Chang, A note on a conjecture of T. H. Foregger, Linear and Multilinear Algebra 15 (1984), 341–344. doi:10.1080/03081088408817602
  4. D. K. Chang, On two permanental conjectures, Linear and Multilinear Algebra 26 (1990), 207–213. doi:10.1080/03081089008817977
  5. K. Melnykova, Notes on Foregger’s conjecture, M.Sc. thesis, University of Manitoba, 2012. mspace.lib.umanitoba.ca/handle/1993/8893
  6. J. Van Name and I. M. Wanless, answers to “On permanent of a square of a doubly stochastic matrix”, MathOverflow, 2022. mathoverflow.net/questions/422029