Some interesting number theory problems

Wenpeng Zhang

REVIEW 4 major objections 5 minor 1 cited by

Some interesting number theory problems

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper derives a universal character-sum identity with gap 2 by equating two published fourth-power-mean formulas, and conjectures a Catalan-number moment formula for cubic exponential sums.

desk verdict A short problems note with one likely-correct character sum identity and a well-motivated conjecture; the derivation is hidden behind a corrupted display and omitted algebra. read the letter →

arxiv 2506.17235 v1 pith:2D264ESM submitted 2025-06-04 math.GM

Wenpeng Zhang This is my paper · ORCID

classification math.GM MSC 11L0311L05

keywords Kloostermansumtwo-termexponentialLegendresymbolfourthpowermeancharacteridentityconjectureasymptoticformula

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's target is a hidden constant in character sums modulo a prime. Taking two published formulas for the fourth power mean of the same exponential sum and equating them, the paper derives the identity that a Legendre-symbol sum over the cubic polynomial $c^3+c^2+c$ differs from a Legendre-symbol sum over $(b^2+1)(b^2+4b+1)$ by exactly $2$, after a factor $(-1/p)$, for every odd prime $p$. The same bridge leads to a conjecture on all even power means of the two-term exponential sum $S(m,1,3;p)$, predicting main term $\frac{1}{k+1}\binom{2k}{k}p^{k+1}$ with error $O(p^{k+1/2})$, verified in the paper for $k=1,\dots,6$ from existing and unpublished results. A sympathetic reader would care because the identity connects two unrelated-looking polynomials through a Kloosterman-sum computation, and the conjecture, if true, would give a complete moment formula for a basic exponential sum.

What carries the argument

The load-bearing object is the fourth power mean of a generalized Kloosterman sum, namely $\sum_{m=0}^{p-1}\left|\sum_{a=1}^{p-1}e((ma^2+a)/p)\right|^4$, because [12] and [13] each produce a closed-form evaluation of it; equating those two expressions, after Legendre-symbol bookkeeping, is what converts the Kloosterman world into an identity between character sums of polynomials. For the conjecture, the working mechanism is the two-term exponential sum $S(m,1,3;p)$ and its even power moments, where the coefficient $\frac{1}{k+1}\binom{2k}{k}$, the Catalan numbers, already appears in the verified low moments.

What would settle it

For $p=211$, outside the paper's reported check, compute both Legendre-symbol sums in (1) directly and compare their difference to $2$; separately evaluate the two fourth-power-mean expressions displayed from [12] and [13] for the same $p$. If the two expressions disagree, or the difference is not $2$, the corollary's deduction fails.

Watch

Extended reading notes

Core claim

The paper asserts that two published evaluations of the same fourth power mean, one due to [12] and one to [13], can be equated to produce a corollary identity: for every odd prime $p$, $$\left(\frac{-1}{p}\right)\sum_{c=1}^{p-1}\left(\frac{$c^{3}$+$c^{2}$+c}{p}\right) - \sum_{b=1}^{p-1}\left(\frac{($b^{2}$+1)($b^{2}$+4b+1)}{p}\right) = 2.$$ The paper does not display the intermediate algebra but reports numerical confirmation for all primes $3 \le p < 200$, and poses as problem (A) the search for a direct elementary proof. The paper also states a conjecture on the moments of the two-term exponential sum $S(m,1,3;p)=\sum_{a=0}^{p-1}e((ma^3+a)/p)$: for every positive integer $k$, $$\sum_{m=0}^{p-1}\left|\sum_{a=0}^{p-1}e\!\left(\frac{$ma^{3}$+a}{p}\right)\right|^{2k} = \frac{1}{k+1}\binom{2k}{k}$p^{{k+1}}$+O\!\left($p^{{k+\frac12}}$\right),$$ and notes that known fourth, sixth and eighth power mean identities make this correct for $k=1,2,3,4$, with an unpublished proof for $k=5,6$.

Load-bearing premise

The corollary rests on the two published fourth-power-mean formulas from [12] and [13] being correct as displayed and on the algebraic manipulation that equates and simplifies them; the paper shows neither the algebra nor a direct proof.

Editorial extensions

If this is right

Identity (1) gives a concrete, quickly testable equality between a cubic and a quartic Legendre-symbol sum, with the constant 2 independent of the prime.
The existence of such a constant opens the classification question (D): whether any two fundamentally different polynomial character sums can differ by a fixed constant other than 2 (or 0).
If the moment conjecture holds, the scaled moments $\frac{1}{p}\sum_{m=0}^{p-1}|S(m,1,3;p)/\sqrt{p}|^{2k}$ approach the Catalan numbers $\frac{1}{k+1}\binom{2k}{k}$, giving a complete asymptotic description of the moment spectrum.
The cases $k=1,\dots,6$ already establish that the conjectured main-term coefficient is not an accident of small exponents, so the remaining obstacle is a proof for general $k$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the Catalan coefficient in the conjectured moment formula is the moment sequence of the semicircle law, so a proof of the conjecture would imply a limiting distribution for the scaled cubic exponential sums, a conclusion the paper does not draw.
Editorial inference: identity (1) may be one member of a larger family; a computer search over small-degree integer polynomials could produce further pairs for problem (B) before a general proof is available.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague, here is my take on Zhang's problems note.

The one thing you should know: the only new result is identity (1), a small but real connection between two character sums, obtained by equating two published fourth-moment formulas. The paper is honest about being a problems list: it proves nothing new beyond that corollary, and even that derivation is not shown. The conjecture on the 2k-th power mean of S(m,1,3;p) is interesting and unifies [16]-[18], but it is conjectural except for k≤6, and the k=5,6 cases rest on an unpublished paper.

What is good: the identity is numerically checked for p<200, the problems (A)-(D) are well-posed and might stimulate a direct elementary proof, and the conjecture is a natural target. Citation pattern is fine; the self-citations are to published peer-reviewed work, not to this preprint.

Soft spots, in proportion: the stress-test note is right. In the display of the formula from [12], the character sum is printed as sum_c ((c+1+c)/p), which is obviously a typesetting error for sum_c ((c^3+c^2+c)/p) (the polynomial that appears in (1)). That makes the reader unable to verify the quoted formula without going to the source. More importantly, the paper says (1) follows 'immediately' from equating [12] and [13], but the algebra is omitted. I checked the signs: with the displayed formulas, the algebra does close, so the risk is whether the quoted formulas themselves are correct. The paper gives no table or code for the numerical check, so the reader cannot reproduce it from the text. These are presentation flaws, not fatal flaws. The main thing missing is a short appendix showing the algebra and a correct rendering of [12].

For whom: this is a note for specialists in elementary/analytic number theory who like character sum identities. It is not a major result, but the identity (1) is the kind of thing that might be worth citing if it is verified. The conjecture could guide future work.

My judgment: as a submitted paper, it is not ready in this form; it needs the derivation and a corrected display. But it should not be desk-rejected: the claim is precise, checkable, and the conjecture is well-motivated. I would send it to a referee, with the request that the referee verify the quoted formulas and the algebra. Given my own skepticism about minor notes, I would not cite it in my own work until the missing details are supplied.

Referee Report

4 major / 5 minor

Summary. The paper is a short research announcement in elementary number theory. Section 1 recalls two published formulas for the fourth power mean of a generalized Kloosterman sum and states that equating them yields Corollary (1), the exact identity (-1/p)∑_{c}((c^3+c^2+c)/p) - ∑_b(((b^2+1)(b^2+4b+1))/p) = 2 for odd primes p. It then poses four open problems (A)-(D) about polynomial character sums with constant difference. Section 2 recalls several known power-mean identities for the cubic exponential sum S(m,1,3;p) and states a conjecture giving the leading term 1/(k+1) C(2k,k) p^{k+1} for its 2k-th moment, asserting that the conjecture is true for k=1,...,4 and that the author proved k=5,6 in an unpublished paper.

Significance. The corollary, if correct, is an elegant and non-obvious relation between two cubic character sums, and it is genuinely parameter-free in the sense that no fitted constants appear. The conjecture is explicit and falsifiable, with a clean Catalan-number coefficient. The paper's strengths are its clarity of motivation and the concrete nature of the proposed problems. Its main weaknesses are that the central deduction is not shown, the quoted formula from [12] is corrupted in the preprint, the empirical verification is not reproducible, and the support for the k=5,6 claim is absent.

major comments (4)

[Section 1, displayed formula from [12]] The character sum in the formula attributed to [12] is typeset as ∑_{c=1}^{p-1} (c+1+c)/p, which is mathematically meaningless. It is almost certainly a rendering error for ∑_{c=1}^{p-1} ((c^3+c^2+c)/p), the same polynomial appearing in (1). As printed, the reader cannot check the formula, and since (1) is derived from it, this is a load-bearing defect.
[Section 1, Eq. (1)] The corollary is asserted to follow 'immediately' from the two quoted formulas, but no algebraic derivation is given. Let A=∑_{c}((c^3+c^2+c)/p) and B=∑_b(((b^2+1)(b^2+4b+1))/p). After correcting the typo above, equating the two formulas for p≡3 mod 4 gives -A-B=2, and equating the p≡1 mod 4 cases gives A-B=2, which is exactly (1). Nevertheless, this calculation should appear in the paper, because the identity is the paper's only new unconditional result and any sign error in the quoted formulas would change the conclusion.
[Section 1, numerical verification] The statement that 'by numerical testing (for all primes 3 ≤ p < 200), the identity (1) is correct' is not accompanied by any table, code, or computational detail. Since this is the only empirical evidence offered for the corollary, the verification should be reproducible from the text; otherwise the reader cannot distinguish a confirmed identity from a plausible coincidence.
[Section 2, Conjecture] The conjecture is stated for all positive integers k, and the paper claims that it is known for k=1,2,3,4 and that the author proved k=5,6 in an unpublished paper. The cited references [16]-[18] only cover the sixth and eighth power means (k=3 and k=4); k=1 is not addressed by any cited result, and k=2 would require [15]. The unpublished claim for k=5,6 is unverifiable. Please either supply the proof or restate these as conjectures.

minor comments (5)

[Section 1, notation] The summation notation 'qX′ a=1' and similar displays are garbled in the typeset; standard \sum' notation would improve readability.
[Section 1, phrase] The phrase 'perhaps the most property is the upper bound estimate' should be rephrased, for example, 'perhaps the most important property is the upper bound estimate'.
[Section 1, Legendre symbol] The Legendre symbol is written inconsistently as both a fraction and as (∗/p); unify the notation throughout.
[Section 2, definition of e(y)] The definition 'e(y) = e2πiy' should be written as e(y)=e^{2π i y} to be mathematically explicit.
[References] Reference [17] is titled 'Acta Mathematics Sinica, Englishe Series'; the correct journal name is 'Acta Mathematica Sinica, English Series'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Corollary (1) follows algebraically from two independent published fourth-moment formulas, and the Section 2 conjecture is an open problem rather than an input-derived claim.

full rationale

The paper's only new unconditional result, Corollary (1), is obtained by equating two displayed formulas for the same fourth power mean of a generalized Kloosterman sum, one quoted from [12] and one from [13]. Both formulas are cited as independent published theorems with stated hypotheses (odd prime p>3), and neither contains the target identity as an assumption; the identity (1) emerges only after algebraic cancellation. This is a deduction from external results, not a prediction fitted to the same data. Although reference [12] is self-authored, it is a parameter-free published result that does not include the target identity among its assumptions, so under the review rules it counts as independent evidence rather than circular support. The Section 2 conjecture is explicitly labeled a 'Conjecture' and is not asserted as a derived theorem; the comments that it is consistent with [16]-[18] and verified for k=5,6 in an unpublished paper are context, not load-bearing steps that presuppose the conjecture. The probable typesetting corruption in the displayed [12] character sum (rendered as (c+1+c)/p instead of the c^3+c^2+c that the algebra requires) and the absence of reproduced algebra and numerical tables are correctness and transparency risks, but they are not circularity. No load-bearing step reduces by definition to its own conclusion.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No new entities are introduced. The paper relies on two published results, one of which is self-authored, plus standard facts about character sums and the Weil bound. The main unstated premise is that the displayed formulas from [12] and [13] are correctly transcribed and algebraic resolution of their equality yields identity (1).

assumptions (3)

domain assumption The fourth-power-mean formulas in [12] and [13] are correct and both evaluate the same quantity.
The corollary (1) is deduced by equating these two published formulas; no proof is reproduced in the present paper.
standard math Standard properties of Legendre symbols and exponential sums over finite fields.
Used throughout Section 1 to manipulate character sums without explicit derivation.
standard math A. Weil's upper bound for two-term exponential sums.
Invoked in Section 2 as the best upper-bound estimate for the sums S(m,n,k;p).

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Cite this review

Pith. "Pith review of Some interesting number theory problems." pith.science (2026). https://pith.science/paper/2D264ESM

@misc{pith2026250617235,
  author       = {Pith},
  title        = {Pith review of: Some interesting number theory problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2D264ESM}},
  note         = {Machine review of arXiv:2506.17235}
}

read the original abstract

The main purpose of this paper is to propose some interesting number theory problems related to the Legendre's symbol and the two-term exponential sums.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On quadratic character sums over quartics
math.NT 2025-07 accept novelty 6.0 of 10

The paper derives a master double-counting formula and several quartic-to-cubic transformation identities for quadratic character sums over finite fields.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · cited by 1 Pith paper

[16]

W. P. Zhang and D. Han, On the sixth power mean of the two-term expo- nential sums, Journal of Number Theory, 136 (2014), 403–413. ?Z?

work page 2014
[18]

T. T. Wang and W. P. Zhang, On the eighth power mean of the two-term exponential sums, Finite Fields and Their Applications, 92 (2023), 102285. 6

work page 2023
[12]

W. P. Zhang, L. Wang and X. G. Liu, One special kind Kloosterman sum and its fourth-power mean, Open Mathematics, 2024, 22: 20240067. ?N?

work page 2024
[13]

S. S. Ning and X. X. Wang, A conversion formula for one kind fourth power mean of Kloosterman sums and character sums modulo p, Journal of Mathematics, 2024, Article ID 9251730. ?W?

work page 2024
[15]

Zhang and W

H. Zhang and W. P. Zhang, The fourth power mean of two-term exponential sums and its application, Mathematical Reports, 19 (2017), 75–81. ?Z?

work page 2017
[1]

H. D. Kloosterman, On the representation of numbers in the form ax2 + by2 + cz2 + dt2, Acta Math., 49 (1926), 407–464. ?I?

work page 1926
[3]

Sali´e, Uber die Kloostermanschen Summen S(u, v; q), Math

H. Sali´e, Uber die Kloostermanschen Summen S(u, v; q), Math. Z., 34 (1931), 91–109. ?Z?

work page 1931
[4]

W. P. Zhang, The fourth and sixth power mean of the classical Kloosterman sums, Journal of Number Theory, 131 (2011), 228–238. ?C?

work page 2011

Show all 17 references

[5]

Chowla, On Kloosterman’s sums, Norkse Vid

S. Chowla, On Kloosterman’s sums, Norkse Vid. Selbsk. Fak. Frondheim, 40 (1967), 70–72. ?E?

1967
[6]

Estermann, On Kloosterman’s sums, Mathematica, 8 (1961), 83–86

T. Estermann, On Kloosterman’s sums, Mathematica, 8 (1961), 83–86. ?A?

1961
[7]

A. V. Malyshev, A generalization of Kloosterman sums and their estimates (in Russian), Vestnik Leningrad Univ., 15 (1960) 59–75. 5 ?Z?

1960
[8]

W. P. Zhang, On the general Kloosterman sums and its fourth power mean, Journal of Number Theory, 104 (2004), 156–161. ?Z?

2004
[9]

W. P. Zhang, On the fourth power mean of the general Kloosterman sums, Indian Journal of Pure and Applied Mathematics, 35 (2004), 237–242. ?Z?

2004
[10]

W. P. Zhang, On the fourth power mean of the general Kloosterman sums, Journal of Number Theory, 169 (2016), 315–326. ?Z?

2016
[11]

W. P. Zhang and S. M. Shen, A note on the fourth power mean of the generalized Kloosterman sums, Journal of Number Theory, 174 (2017), 419–

2017
[14]

Weil, On some exponential sums, Proceedings of the National Academy of Sciences of the United States of America, 34 (1948), 203–210

A. Weil, On some exponential sums, Proceedings of the National Academy of Sciences of the United States of America, 34 (1948), 203–210. ?Z?

1948
[17]

W. P. Zhang and Y. Y. Meng, On the sixth power mean of the two-term exponential sums, Acta Mathematics Sinica, Englishe Series, 38 (2022), 510–

2022

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [16]

W. P. Zhang and D. Han, On the sixth power mean of the two-term expo- nential sums, Journal of Number Theory, 136 (2014), 403–413. ?Z?

work page 2014

[2] [18]

T. T. Wang and W. P. Zhang, On the eighth power mean of the two-term exponential sums, Finite Fields and Their Applications, 92 (2023), 102285. 6

work page 2023

[3] [12]

W. P. Zhang, L. Wang and X. G. Liu, One special kind Kloosterman sum and its fourth-power mean, Open Mathematics, 2024, 22: 20240067. ?N?

work page 2024

[4] [13]

S. S. Ning and X. X. Wang, A conversion formula for one kind fourth power mean of Kloosterman sums and character sums modulo p, Journal of Mathematics, 2024, Article ID 9251730. ?W?

work page 2024

[5] [15]

Zhang and W

H. Zhang and W. P. Zhang, The fourth power mean of two-term exponential sums and its application, Mathematical Reports, 19 (2017), 75–81. ?Z?

work page 2017

[6] [1]

H. D. Kloosterman, On the representation of numbers in the form ax2 + by2 + cz2 + dt2, Acta Math., 49 (1926), 407–464. ?I?

work page 1926

[7] [3]

Sali´e, Uber die Kloostermanschen Summen S(u, v; q), Math

H. Sali´e, Uber die Kloostermanschen Summen S(u, v; q), Math. Z., 34 (1931), 91–109. ?Z?

work page 1931

[8] [4]

W. P. Zhang, The fourth and sixth power mean of the classical Kloosterman sums, Journal of Number Theory, 131 (2011), 228–238. ?C?

work page 2011