bibkey: zhang2025problems authors: Wenpeng Zhang year: 2025 title: “Some interesting number theory problems” doi: 10.48550/arXiv.2506.17235 url: https://arxiv.org/abs/2506.17235v1 claim: “Question (D) asks whether the fixed constant difference c in identity (2), for fundamentally different integer-coefficient polynomials, can only be 0 or 2.” strata_touched:

• D5/S3/ArithSums/ZhangCharacterSumDifferenceRefutation license: citation-only triage: anchor

Zhang’s polynomial character-sum questions

Verified locator

• DOI: 10.48550/arXiv.2506.17235
• URL: https://arxiv.org/abs/2506.17235v1

The arXiv record has only version 1. The Legendre symbol is introduced on printed page 2, and the numbered questions occur on printed page 3. No journal publication or journal DOI is asserted here.

Source statement

After a corollary stated for every odd prime, the paper asks:

(B). Whether there are infinitely many pairs of fundamentally different integer coefficients polynomials f(x) and g(x) (That is, (f(x)/p) != (g(x)/p)) such that sum from x=1 to p-1 of (f(x)/p) minus sum from x=1 to p-1 of (g(x)/p) = c, (2) where c is a fixed constant.

(C). What are the common properties between the polynomials f(x) and g(x) that satisfy identity (2).

(D). Whether the values of c can only be 0 or 2?

Here (a/p) is the Legendre symbol modulo p. Identity (2) does not print an explicit universal binder for p. The formal statement reads it as one fixed c valid for all odd primes, following the preceding corollary’s quantifier. The counterexample also satisfies readings restricted to primes greater than three and readings requiring the two symbol functions to differ at only some prime.

Bounded prior-resolution evidence

Nica, On quadratic character sums over quartics, arXiv:2507.09991, gives a direct proof of the paper’s identity (1), answering Question (A), but does not discuss Question (D). Exact identifier, author, title, quoted-question, arXiv, Semantic Scholar, and public repository searches located no prior resolution of Question (D) in the searched scope. Semantic Scholar returned no citing records. Google Scholar was not verified.

The literature search is bounded and does not establish a worldwide priority claim. Its completeness remains ASSUMED-UNVERIFIED.