A Constant-One Difference of Polynomial Character Sums

Abstract

Two quadratic-polynomial Legendre sums differ by one for every odd prime.

The Legendre symbol notation and the range from one through p-1 are those of identity (2). The paper’s preceding corollary quantifies over odd primes.

Definition 1.1 (The polynomial Legendre character sum).

$$\forall p\in \mathbb{N},(\mathrm{Prime}\left(p\right))\Rightarrow (\forall f\in \mathbb{Z}[X],\mathrm{characterSum}\left(p,f\right)=\sum _{x=1}^{p-1}(\frac{f\left((x:\mathbb{Z})\right)}{p}))$$

Formalization. D5/S3/ArithSums/ZhangCharacterSumDifferenceRefutation.characterSum (✓ std3).

Citation. Wenpeng Zhang (2025). Some interesting number theory problems. DOI: 10.48550/arXiv.2506.17235. URL: https://arxiv.org/abs/2506.17235v1.

Commentary.

For a prime p and an integer-coefficient polynomial f, this is the sum of the Legendre symbols of f(x) over the integers from one through p-1.

Definition 1.2 (Fundamental difference on the summation domain).

$$\forall p\in \mathbb{N},(\mathrm{Prime}\left(p\right))\Rightarrow (\forall f\in \mathbb{Z}[X],g\in \mathbb{Z}[X],(\mathrm{FundamentallyDifferent}\left(p,f,g\right))\iff (\exists x\in \mathbb{N},((1\le x)\wedge (x<p))\wedge ((\frac{f\left((x:\mathbb{Z})\right)}{p})\ne (\frac{g\left((x:\mathbb{Z})\right)}{p}))))$$

Formalization. D5/S3/ArithSums/ZhangCharacterSumDifferenceRefutation.FundamentallyDifferent (✓ std3).

Citation. Wenpeng Zhang (2025). Some interesting number theory problems. DOI: 10.48550/arXiv.2506.17235. URL: https://arxiv.org/abs/2506.17235v1.

Commentary.

The two symbol-valued functions are different when some x in the same finite summation domain gives unequal values. This is equivalent to inequality of those functions on that domain.

Definition 1.3 (The constant-value assertion in Question (D)).

$$(claim)\iff (\forall f\in \mathbb{Z}[X],g\in \mathbb{Z}[X],c\in \mathbb{Z},(\forall p\in \mathbb{N},((\mathrm{Prime}\left(p\right))\wedge (p\ne 2))\Rightarrow ((\mathrm{FundamentallyDifferent}\left(p,f,g\right))\wedge (\mathrm{characterSum}\left(p,f\right)-\mathrm{characterSum}\left(p,g\right)=c)))\Rightarrow ((c=0)\vee (c=2)))$$

Formalization. D5/S3/ArithSums/ZhangCharacterSumDifferenceRefutation.claim (✓ std3).

Citation. Wenpeng Zhang (2025). Some interesting number theory problems. DOI: 10.48550/arXiv.2506.17235. URL: https://arxiv.org/abs/2506.17235v1.

Commentary.

Question (B) asks: “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 Σ_{x=1}^{p−1} (f(x)/p) − Σ_{x=1}^{p−1} (g(x)/p) = c, (2) where c is a fixed constant.” Question (D) asks: “Whether the values of c can only be 0 or 2?” Identity (2) does not print an explicit universal binder for p. Here one integer c is bound outside the universal odd-prime condition, following the preceding corollary. Thus the same c is the difference for every odd prime, and the two symbol functions differ at every such prime.

Theorem 1.4 (A constant-one counterexample).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S3/ArithSums/ZhangCharacterSumDifferenceRefutation.result (✓ std3). ∎

Resolves. Problems/zhang-character-sum-difference-question-d-refutation (refuted) by D5/S3/ArithSums/ZhangCharacterSumDifferenceRefutation.result.

Source. Repository-derived.

Acknowledgement. Wenpeng Zhang (2025). Some interesting number theory problems. DOI: 10.48550/arXiv.2506.17235. URL: https://arxiv.org/abs/2506.17235v1.

Commentary.

Take f(X)=X^2 and g(X)=(X+1)^2. For every odd prime p, the first sum is p-1. The second is p-2 because its final term, at x=p-1, is zero and every earlier term is one. At that same endpoint the first symbol is one, so the functions differ. Their sum difference is the fixed integer c=1, which is neither zero nor two.

References

• Truth anchor: D5/S3/ArithSums/ZhangCharacterSumDifferenceRefutation.FundamentallyDifferent
• Truth anchor: D5/S3/ArithSums/ZhangCharacterSumDifferenceRefutation.characterSum
• Truth anchor: D5/S3/ArithSums/ZhangCharacterSumDifferenceRefutation.claim
• Truth anchor: D5/S3/ArithSums/ZhangCharacterSumDifferenceRefutation.result