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Two improved lower bounds for the 2-adic valuation of the constant of the Prouhet-Tarry-Escott problem

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

For an ideal solution of the Prouhet--Tarry--Escott problem of size nn the associated constant CC is divisible by a large power of 22. Restricting to symmetric solutions, we prove two new lower bounds for v2(C)v_2(C): v2(C)≥14v_2(C)\ge14 at n=15n=15, improving 1212, and v2(C)≥19v_2(C)\ge19 at n=20n=20, improving 1818; both earlier bounds are those of the companion article hexagon:2611.00057. The first rests on two identities satisfied by every symmetric solution of odd size, one relating v2(C)v_2(C) to a difference of two products and one turning the index relation into an equality, together with an arithmetic criterion for a block of entries of equal valuation. The second rests on an identity internal to each of the two sides, from which we deduce that the vanishing of the cross invariant RR forces k0≡0(mod4)k_0\equiv0\pmod4 when k0≥4k_0\ge4. Two self-contained verifiers accompany the note.

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by independent AI reviews and by exact computation. No professional mathematician has reviewed the note yet.

Tools used

Anthropic
Claude Opus

References

  1. R. Bera and S. G. Nair, Newton polygons and the constant associated with the Prouhet--Tarry--Escott problem , Acta Arith. 213 (2024), 75--95.
  2. D. Coppersmith, M. J. Mossinghoff, D. Scheinerman and J. M. VanderKam, Ideal solutions in the Prouhet--Tarry--Escott problem , preprint, arXiv:2304.11254.arXiv
  3. M. Filaseta and M. Markovich, Newton polygons and the Prouhet--Tarry--Escott problem , J. Number Theory 174 (2017), 384--400.
  4. J. C. Foster, Polynomials, primes and the PTE problem , Ph.D. thesis, University of South Carolina, 2021.
  5. H. Kleiman, A note on the Tarry--Escott problem , J. Reine Angew. Math. 278/279 (1975), 48--51.
  6. D. Niedbala Giraudin, The 22-adic valuation of the constant for symmetric ideal Prouhet--Tarry--Escott solutions , preprint, hexagon:2611.00057 (2026).hexagon

Version history

  1. v1Submitted by David Niedbala GiraudinInitial depositCurrentNov 03, 2026