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The 2-adic valuation of the constant for symmetric ideal Prouhet-Tarry-Escott solutions

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, let CC denote the associated constant, and let Cn‾\overline{C_n} be the greatest common divisor of these constants. Filaseta and Markovich obtained v2(C8‾)≥6v_2(\overline{C_8})\ge6 and determined v2(C9‾)=9v_2(\overline{C_9})=9, and asked for improved lower bounds in the range 10≤n≤1210\le n\le12; Bera and Nair gave bounds for n=10n=10 and n=12n=12. We treat the symmetric case at every size, even and odd, by 22-adic arguments throughout. A single identity, the row law, governs both parities. For nn even, over Z\mathbb{Z}, it splits into two regular systems and yields v2(C)≥6,8,11,12,12,16,18,18v_2(C)\ge6,8,11,12,12,16,18,18 for n=6,8,10,…,20n=6,8,10,\dots,20. The values at n=6,8,10,12n=6,8,10,12 are attained and hence exact for the symmetric constant: in the symmetric case this closes both brackets of Bera--Nair, and gives v2(C8′)=8v_2(C'_8)=8, for which the published information was v2(C8‾)≥6v_2(\overline{C_8})\ge6. We also record that the two lower bounds of Bera--Nair, 99 at n=10n=10 and 1111 at n=12n=12, are attained by 22-adic solutions symmetric about −1/2-1/2, so that no argument using 22-adic information alone can raise them. For nn odd, over Z2\mathbb{Z}_2, it combines with a finite-difference lemma refining a congruence of Filaseta--Markovich to give v2(C)≥11,12,12v_2(C)\ge11,12,12 for n=11,13,15n=11,13,15, against 88, 1010 and 1111 from Kleiman's bound (n−1)!∣C(n-1)!\mid C; the chain also reproduces the known values 4,6,94,6,9 at n=5,7,9n=5,7,9. At n=11n=11 the bound is sharp: the two quadratic near-solutions of Coppersmith, Mossinghoff, Scheinerman and VanderKam are symmetric ideal solutions over Z2\mathbb{Z}_2 with v2(C)=11v_2(C)=11, since in both of the real quadratic fields involved the prime 22 splits. This answers, for n=11n=11 and p=2p=2, two questions of Filaseta--Markovich, and exhibits for those sizes the case p≤np\le n of the question closing Rees--Smyth, who construct pp-adic ideal solutions only for p>np>n: at the prime 22 there is no local obstruction at these sizes. At n=13n=13 we exhibit a symmetric ideal solution over Z2\mathbb{Z}_2, obtained from a second object of the same authors, although its constant has valuation 1717.

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

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Show all 16 referencesShow fewer references
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Version history

  1. v1Submitted by David Niedbala GiraudinInitial depositCurrentNov 03, 2026