bibkey: kotesovec2024a368959
authors: Václav Kotěšovec
year: 2024
title: "OEIS A368959: nonzero complete monomial exponential sums"
doi: null
url: https://oeis.org/A368959
claim: "Conjecture 1: This is a subset of A001694. Conjecture 2: This sequence contains all the terms from A338325, plus 243, 972, 1944, 2401, ..."
strata_touched:
  - D5/S3/Arith/CharacterSums/KotesovecConjectureOne
  - D5/S3/Arith/CharacterSums/KotesovecConjectureTwo
license: citation-only
triage: anchor

Complete monomial exponential sums

Verified locator

Source: https://oeis.org/A368959

The NAME field states:

Numbers m such that Sum_{k=0..m-1} exp(2Pii*k^s/m) <> 0 for all s>=2.

The COMMENTS field states:

Conjecture 1: This is a subset of A001694.

Conjecture 2: This sequence contains all the terms from A338325, plus 243, 972, 1944, 2401, …

The NAME field of https://oeis.org/A338325 states:

Biquadratefree powerful numbers: numbers whose exponents in their prime factorization are either 2 or 3.

These are OEIS field locators; the source has no page numbers. A001694 denotes powerful numbers. Membership is restricted to positive integers. The ellipsis in Conjecture 2 supplies no further quantified list beyond the four named terms.

Mathematical scope

A prime dividing the modulus exactly once forces vanishing at that prime degree. For a prime power p^(a+1), rational cyclotomic divisibility forces equality of the coefficients at r and r+p^a for r<p^a in the power-map count polynomial. Unequal fiber counts give nonvanishing. Prime squares, prime cubes, 243 and 2401 satisfy the resulting criterion; coprime products supply the family and the products 972 and 1944.

Computational comparison

The experiment is in the private repository the-omega-institute/trureturing-experiments, commit 69e451300aca2a3ed4ebf255d5b3adceaaca1c14, under docs/reports/kotesovec-2024-a368959-monomial-sums. Its check.py has SHA256 d6dc43174ffdd445f069c7b355eeb596d3066c663e08819e20cff2d23e4064d9. The checker exits 0 with RESULT PASS; this is computational comparison rather than a premise of the Lean proofs.

Literature boundary

The bounded screen recorded in issues #19241 and #19242 covers the OEIS entries, Fried arXiv:2607.24832, Lam–Leung arXiv:math/9511209, Cochrane–Zheng’s exponential-sum work, Pigno’s dissertation and MathOverflow 480590. Exhaustive prior-art clearance is ASSUMED-UNVERIFIED. The final Pigno–Pinner journal version and Liu–Liu, Truncated polynomials with restricted digits, remain unverified. No mathematical originality claim is made.