REVIEW 3 major objections 3 minor 2 references
The distribution of powers of primes related to the Frobenius problem
T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read This paper proves that, for any fixed $k$, the number of prime powers $p^k$ up to the Frobenius gap $cd-c-d$ representable as $cx+dy$ is asymptotic to $\frac{k}{k+1}\frac{g^{1/k}}{\log g}$ as $c\to\infty$.
desk verdict A natural extension of DZZ to prime powers, but the major-arc error estimates don't close and the Stieltjes integration has a sign error; the main result is plausible but not proved as written. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The engine is the pair of exponential sums $f(\alpha)=\sum_{n^k\le g}\Lambda(n)e(\alpha n^k)$ and $h(\alpha)=\sum_{0\le x\le d,\,0\le y\le c}e(\alpha(cx+dy))$; orthogonality makes the weighted count $\psi_{c,d}$ equal to $\int_0^1 f(\alpha)h(-\alpha)\,d\alpha$. On the major arcs, $f$ is replaced by the Gauss-sum approximation $S(q,a)v(\beta)$, where $S(q,a)=q^{-1}\sum_{n=1}^q e(an^k/q)$ and $v(\beta)=\frac1k\sum_{n\le g} n^{1/k-1}e(\beta n)$, and the resulting main term is evaluated through an exact lattice-point count of representable integers. The minor arcs are controlled by a Weyl-sum estimate over primes, and the exceptional small-$c$ range is handled with the Siegel-Walfisz theorem and the prime number theorem.
What would settle it
Take k=2 and d=c+1, so g=$c^{2}$-1, and let c run through a growing sequence such as powers of 2. For each c, compute exactly the number N of integers n<c for which $n^{2}$ is representable as cx+(c+1)y with x,y\ge 0; the theorem predicts N/$g^{{1/2}}$\to 1/3. If the ratio does not approach 1/3, the central claim is false.
Extended reading notes
Core claim
On the paper's own terms, the central discovery is Theorem 1.1: for coprime $1<c<d$ with $c$ sufficiently large, the count $\pi_{c,d,k}$ of prime powers $p^k\le g=cd-c-d$ of the form $cx+dy$ satisfies $\pi_{c,d,k}\sim \frac{k}{k+1}\frac{g^{1/k}}{\log g}$. The proof is built from two intermediate asymptotics: the unweighted count $N$ of all $k$-th powers $n^k\le g$ representable as $cx+dy$ satisfies $N\sim g^{1/k}/(k+1)$, and the von Mangoldt weighted count $\psi_{c,d}$ over such prime powers satisfies the same asymptotic. The step from $\psi$ to $\pi$ is a partial-summation argument using Chebyshev estimates, so the arithmetic content is concentrated in the asymptotic for $N$, obtained by a major-arc/minor-arc decomposition of the circle-method integral.
Load-bearing premise
The whole result rests on the assumption that the contribution from non-principal major arcs—those centred at rational points with denominator between 2 and Q=(log g)^m—is negligible; the written bound for that contribution is of size $dQ^{3}$, which is not small compared with the claimed main term for the full range of d allowed by the hypotheses, so an unproved cancellation is needed there.
Editorial extensions
If this is right
- For every fixed $k$, the fraction of prime powers $p^k\le g$ representable as $cx+dy$ tends to $1/(k+1)$ as $c\to\infty$; in particular, about one third of prime squares and one quarter of prime cubes are representable in the limit.
- The unweighted count $N$ of representable $k$-th powers is asymptotic to $g^{1/k}/(k+1)$, refining the classical half-density statement for all integers up to $g$ when restricted to the $k$-th power subsequence.
- The weighted and unweighted counts have the same leading term, so the logarithmic weights introduced by $\Lambda$ do not distort the asymptotic.
- Because the transition from $\psi$ to $\pi$ is elementary partial summation, any improvement in the major/minor arc estimates transfers directly to the prime-power count.
Reading between the lines
- This suggests a broader counting principle: for any sparse sequence whose exponential sum has a major-arc approximation, the count of its representable members below $g$ should factor as the sequence's total count times a local density; here that factor is $1/(k+1)$.
- One testable extension is to fix $k=1$ and $k=2$ for the same pair $(c,d)$: the theorem predicts both limits simultaneously, and for $d=c+1$ exact counts can be computed for moderately large $c$ to check the two constants.
- The argument also points toward a uniformity statement beyond the written error terms: the ratio $\pi_{c,d,k}/(g^{1/k}/\log g)$ should approach $k/(k+1)$ independently of how $d$ varies with $c$, which is stronger than what the explicit bounds show.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves an asymptotic formula for the number of prime powers p^k ≤ g_{c,d} that can be written as cx+dy with x,y nonnegative integers, where g_{c,d}=cd−c−d and gcd(c,d)=1. The main theorem (Theorem 1.1) states that, for fixed k≥1 and as c→∞, this count is asymptotic to k/(k+1) · g^{1/k}/log g. The proof follows the Hardy–Littlewood method, using major and minor arc estimates; the major arcs are analyzed via a transition to the unweighted counting function F, and the minor arcs via exponential sum bounds of Kumchev and a lemma of Ding–Zhai–Zhao. The paper also proves auxiliary results for the weighted von Mangoldt sum ψ_{c,d} (Theorem 1.3) and for the total number N of representable k-th powers (Theorem 1.2).
Significance. If correct, the result is a natural and nontrivial extension of the k=1 theorem of Ding–Zhai–Zhao, giving a quantitative density for representable prime powers: asymptotically 1/(k+1) of all prime powers up to g are representable. The main constant agrees with a simple lattice-point heuristic, and the paper relies on standard tools (PNT, Siegel–Walfisz, Kumchev's estimates, Vaughan's method) without fitted parameters, which lends plausibility. However, a load-bearing error term in the major arc treatment is not shown to vanish at the required rate, so the main theorems are not established as written.
major comments (3)
- [§4, Lemma 4.3 and Proposition 4.1] The contribution of the nonprincipal major arcs (2≤q≤Q) is bounded using |F(α)|≤g, but in fact |F(α)|≤⌊g^{1/k}⌋+1. The resulting error is dQ^3 as written, and even after the correction it becomes dQ^3 g^{1/k−1}. With the paper's choice Q=(log g)^m and c≥(log g)^{m+1}, and using d/g≈1/c, this corrected term is g^{1/k}Q^3/c = g^{1/k}(log g)^{2m−1}, which is not o(g^{1/k}/log g) since m≥10. Thus the error term in Proposition 4.1 is not negligible, and the proposition is not proved.
- [§4, proof of Theorem 1.2, Eq. (4.7)] The same problem appears in the proof of Theorem 1.2, where the nonprincipal major arcs are bounded by dQ^3 g^{1/k−1}. With Q=(log g)^m and c≥(log g)^{m+1}, this term is of size g^{1/k}(log g)^{2m−1}, which is larger than the claimed main term g^{1/k}/(k+1) by a power of log g. Consequently, the asymptotic N∼g^{1/k}/(k+1) is not derived from the written estimates.
- [§5, Eq. (5.1) and Theorem 1.3] Equation (5.1) asserts that ψ_{c,d}=N+O(g^{1/k}/log g) by combining Proposition 3.1 and Proposition 4.1. Since Proposition 4.1 contains the unproved and in fact too-large dQ^3 (or dQ^3 g^{1/k−1}) term, the estimate (5.1) is unjustified. As Theorem 1.3 and Theorem 1.1 rely on this estimate, the main result is not established. The proof would need an additional cancellation mechanism, for instance from the Gauss sums S(q,a) or from an average bound on h over a, to make the nonprincipal major arcs negligible.
minor comments (3)
- [§5, Lemma 5.1 proof] The summation index in the Euler–Maclaurin formula is written as '0<k≤l', but k is already the fixed exponent in the paper; this should be a different letter such as m. Also, the statement says κ2 and κ3 are 'positive integers' but they appear as constants in exponential decay; they should be 'positive constants'.
- [Throughout] The name 'Ska/suppress lba' appears in the introduction and references; this is a corrupted rendering of 'Skałba' and should be fixed.
- [Various] There are several typographical issues: the title contains 'RELA TED'; 'comes form' in the proof of Proposition 4.1 should be 'comes from'; 'Sylverter' in the proof of Theorem 1.2 should be 'Sylvester'. These do not affect the mathematics but should be corrected.
Circularity Check
No circularity: the proof is a self-contained Hardy–Littlewood argument built on external theorems, with no fitted parameters and no self-citation chain.
full rationale
The paper's derivation chain is not circular. The main asymptotic for π_{c,d} is obtained from Theorem 1.3 via standard partial summation, and Theorem 1.3 is obtained by combining the major-arc Proposition 4.1 and the minor-arc Proposition 3.1 with the prime number theorem and Siegel–Walfisz. The intermediate Theorem 1.2 for N is proved independently by the circle method using Vaughan's theorem [Vau97] and Kumchev's exponential sum estimates [Kum06], with no step assuming the target asymptotic. The only cited lemma taken verbatim from prior work is Lemma 3.2 of [DZZ], a bound on ∫|h(−α)|dα; this is an external result by different authors, not a self-citation, and it is used only as an auxiliary estimate. No parameter is fitted to data and no prediction is renamed from an input. The skeptical concern about the size of nontrivial major-arc errors (Lemma 4.2–4.3 and the proof of Theorem 1.2) is a question about whether the stated estimates are strong enough to yield the claimed error term; it is a correctness or gap issue, not a circularity issue. Accordingly, the circularity score is 0.
Assumptions & free parameters
assumptions (6)
- standard math Prime number theorem in the form ψ(x)=x+O(x exp(-κ√log x))
- standard math Siegel-Walfisz theorem for primes in arithmetic progressions
- standard math Kumchev's Theorem 3, Weyl sums over primes
- standard math Vaughan's Theorem 4.1: asymptotic for exponential sums of k-th powers
- domain assumption Lemma 3.2 of Ding-Zhai-Zhao: ∫_0^1 |h(-α)| dα ≪ (log g)^2
- standard math Sylvester's exact symmetry: exactly one of m and g-m is representable, and K_g=(g+1)/2
Cite this review
Pith. "Pith review of The distribution of powers of primes related to the Frobenius problem." pith.science (2026). https://pith.science/paper/UH2CDFXE
@misc{pith2026241218898,
author = {Pith},
title = {Pith review of: The distribution of powers of primes related to the Frobenius problem},
year = {2026},
howpublished = {\url{https://pith.science/paper/UH2CDFXE}},
note = {Machine review of arXiv:2412.18898}
}
abstract
Let $1<c<d$ be two relatively prime integers, $g_{c,d}=cd-c-d$ and $\mathbb{P}$ is the set of primes. For any given integer $k \geq 1$, we prove that $$\#\left\{p^k\le g_{c,d}:p\in \mathbb{P}, ~p^k=cx+dy,~x,y\in \mathbb{Z}_{\geqslant0} \right\}\sim \frac{k}{k+1}\frac{g^{1/k}}{\log g} \quad (\text{as}~c\rightarrow\infty),$$ which gives an extension of a recent result of Ding, Zhai and Zhao.
Reference graph
Works this paper leans on
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[RAS20] J. L. Ram´ ırez Alfons´ ın and M. Ska/suppress lba,Primes in numerical semigroups, C. R. Math. Acad. Sci. Paris 358 (2020), no. 9-10, 1001–1004, DOI 10.5802/crmath.104. [Syl82] J. J. Sylvester, On Subvariants, i.e. Semi-Invariants to Binary Quantics of an Unlimited Order, Amer. J. Math. 5 (1882), no. 1-4, 79–136, DOI 10.2307/2369536. [Vau97] R. C....
Reviewed August 11, 2026 · model on record in the stance chip above.
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