{"id":"1284eb84-a2ca-4547-a49e-c340b97d22e9","arxiv_id":"2412.18898","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The number of prime powers p^k below cd-c-d that can be written as cx+dy is asymptotically k/(k+1) times g^{1/k}/log g as c grows.","lead":"Two coprime coin denominations c and d leave a largest unpayable amount cd-c-d. This paper counts how many prime powers below that bound are payable, and finds the exact asymptotic formula as c grows. The result extends a recently solved conjecture about ordinary primes to every fixed power, sharpening the theory of the Frobenius coin problem.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Nontrivial major-arc error bounds are too weak: with Q=(log g)^m and c≥(log g)^{m+1}, the crude h≪qd estimate leaves an error of size g^{1/k}(log g)^{2m-1}, which is not o(g^{1/k}/log g).","rationale":"The reader's weakest assumption correctly identifies the nonprincipal major arcs as the load-bearing gap. My independent reading confirms that the crude bound h≪qd is the only estimate used there, and the arithmetic of the chosen Q and c leaves an error exceeding the claimed main term. This is not merely a matter of choosing a larger Q or a larger lower bound for c, because Q is already a power of log g and c is required only to exceed (log g)^{m+1}; increasing m worsens the ratio Q^3/c. A fix would require an additional cancellation estimate, most plausibly from the Gauss sums S(q,a) or from a sharper average of h over the reduced residues a. The underlying main constant is supported by a separate lattice-point heuristic, and the general method of Ding–Zhai–Zhao is credible, so the appropriate verdict remains conditional rather than rejection or acceptance. I agree with the reader's assessment and recommend no change to the verdict.","tokens_in":11251,"tokens_out":7018,"duration_ms":66429,"concrete_test":"Re-derive the nontrivial major-arc sum in Proposition 4.1 exactly: compute T = Σ_{q=2}^{Q} Σ_{(a,q)=1} S(q,a) ∫_{−Q/(qg)}^{Q/(qg)} v(β) h(−a/q−β) dβ, with Q=(log g)^m and c=(log g)^{m+1}, d chosen so g=cd−c−d. Use the explicit forms of S(q,a), v, and h to evaluate the inner a-sum (or bound it nontrivially via Gauss-sum cancellation). If the resulting bound is not o(g^{1/k}/log g), the proof of Theorem 1.2 fails at this point; if a saving of at least (log g)^{2m} emerges, the gap is closable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central asymptotic depends on Proposition 4.1 and Theorem 1.2, but the contribution of the nonprincipal major arcs (2≤q≤Q) is not shown to be small under the stated parameter choices. In Lemma 4.3 and the proof of Theorem 1.2, the nontrivial arcs are bounded using only |h(−a/q−β)|≪qd and |F|≤g (or g^{1/k} for v), yielding errors dQ^3 or dQ^3 g^{1/k−1}. With Q=(log g)^m and the paper's condition c≥(log g)^{m+1}, and using d/g≈1/c, even the more favorable bound is dQ^3 g^{1/k−1} = (d/g)g^{1/k}Q^3 ≈ g^{1/k} Q^3/c = g^{1/k}(log g)^{3m−(m+1)} = g^{1/k}(log g)^{2m−1}. Since m=max{3·2^{k−1},⌈2δ+8⌉}+1≥10, this error is larger than the claimed main term g^{1/k}/log g by a factor (log g)^{2m}. The written proof supplies no cancellation from the Gauss sums S(q,a) or from averaging h over a, so the error term in Proposition 4.1 is not o(g^{1/k}/log g); consequently Equation (5.1), Theorem 1.3, and Theorem 1.1 are not established as written. A smaller separate issue is that Lemma 4.3's dQ^3 term appears to use |F|≤g instead of |F|≤g^{1/k}, a factor of g^{1−1/k} larger, but removing this extra factor does not repair the logarithmic gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":11641,"tokens_out":10109,"duration_ms":82298,"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":[{"comment":"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.","section":"§4, Lemma 4.3 and Proposition 4.1"},{"comment":"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.","section":"§4, proof of Theorem 1.2, Eq. (4.7)"},{"comment":"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.","section":"§5, Eq. (5.1) and Theorem 1.3"}],"minor_comments":[{"comment":"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'.","section":"§5, Lemma 5.1 proof"},{"comment":"The name 'Ska/suppress lba' appears in the introduction and references; this is a corrupted rendering of 'Skałba' and should be fixed.","section":"Throughout"},{"comment":"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.","section":"Various"}],"recommendation":"major_revision","confidential_remarks":"The paper is a straightforward extension of the DZZ method to prime powers, and the main constant is plausible. However, the referee is in agreement with the stress-test concern: the nonprincipal major arcs are estimated too crudely, and the resulting error term is far larger than the claimed main term under the parameter choices in Section 5. This is a genuine gap in the proof of Propositions 4.1 and Theorem 1.2, and it propagates to the main theorem. The gap may be fixable with a deeper analysis of the Gauss sums or an averaging argument, but as written the manuscript does not establish the stated results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper extends Ding–Zhai–Zhao's resolution of the Ramirez Alfonsin–Skalba conjecture from primes to k-th powers of primes, for every fixed k. The main theorem is the expected asymptotic: the count of representable prime powers p^k ≤ g is ~ k/(k+1) g^{1/k}/log g. That's a natural, worthwhile extension, and the heuristic constant is right: the smooth-weight computation with K_t ~ t^2/(2cd) gives N ~ g^{1/k}/(k+1), so the target theorem is almost certainly true. The paper is honest in its dependencies: PNT, Siegel-Walfisz, Kumchev's estimates, Vaughan, and a lemma from DZZ, with no fitted parameters.\n\nThe problem is the proof as written. The nontrivial major arcs (2≤q≤Q) are bounded by absolute values only: in Lemma 4.3 the contribution is bounded by dQ^3, and in Theorem 1.2 by dQ^3 g^{1/k-1}. With Q=(log g)^m and c≥(log g)^{m+1}, these are of size g^{1/k}(log g)^{2m-1}, far larger than the claimed main term g^{1/k}/log g. Using |F|≤g in Lemma 4.3 is also a typo (should be g^{1/k}) but correcting it doesn't fix the logarithmic gap. The proof needs real cancellation, e.g. averaging the Gauss sums S(q,a) or a sharper bound on h over a, and none is supplied. The stress-test note is correct on this.\n\nSecondary issue: the integration by parts in Theorem 1.2 has a sign error. They write −(1/k)(1−1/k)∫ K_t t^{1/k−2}dt, but the correct sign is plus. With the displayed minus, the boundary and integral terms would not combine to 1/(k+1); they would give 1/(k(k+1)). This may be a typo, but as written the computation doesn't justify the main constant.\n\nSo the central claim is plausible and likely correct, but the submitted proof has a load-bearing gap. A serious referee could probably help the authors close it: the missing ingredient is standard in the circle method, though not trivial. I'd send it to review, with the expectation of major revision. It's not ready to accept as is.\n\nWho's this for? Anyone working on Frobenius problems or the circle method applied to numerical semigroups. The result is a clean, if expected, extension of DZZ.","headline":"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.","tokens_in":12175,"tokens_out":12005,"would_cite":false,"duration_ms":95732,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B25","11F11","11F30","11L05","11N37","11T23"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $c\\to\\infty$.","keywords":["Frobenius problem","prime powers","Hardy-Littlewood method","exponential sums","Siegel-Walfisz theorem","numerical semigroups","asymptotic density"],"falsifier":"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.","tokens_in":11041,"feed_emoji":"🔢","tokens_out":15207,"duration_ms":131183,"temperature":0.7,"pith_summary":"The two-coin Frobenius problem asks which integers can be written as $cx+dy$ with $x,y\\ge 0$, for coprime $1<c<d$; the largest unreachable integer is $g=cd-c-d$. This paper studies the sparse subsequence consisting of powers of primes, and its main theorem pins down the count as $c$ grows. For every fixed $k\\ge 1$, the number of prime powers $p^k\\le g$ that are representable is asymptotic to $\\frac{k}{k+1}\\frac{g^{1/k}}{\\log g}$, which is exactly $1/(k+1)$ of all prime powers up to $g$. The result extends the previously known case $k=1$ and gives the first-order answer for all exponents, with the same Hardy-Littlewood mechanism used in the prime case.","feed_headline":"Representable prime powers: exactly one in k+1","feed_subtitle":"Asymptotically, exactly one in k+1 prime powers up to cd-c-d is representable as cx+dy.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the k=1 case of the conjecture and the general circle-method strategy, including the L1 bound for h used in the minor-arc estimate.","marker":"[DZZ]"},{"why":"Provides the Weyl-sum estimate over primes that controls f on the minor arcs.","marker":"[Kum06]"},{"why":"Supplies the Hardy-Littlewood machinery: the approximation F=S(q,a)v(beta)+... and the bounds on v(beta).","marker":"[Vau97]"},{"why":"Formulates the conjecture that the present theorem extends to prime powers.","marker":"[RAS20]"},{"why":"Gives the Frobenius number cd-c-d and the exact half-count of representable integers used to evaluate the main term.","marker":"[Syl82]"},{"why":"Provides the exponential-sum estimate for F(alpha) used to bound the minor-arc contribution in Lemma 4.3.","marker":"[LZ23]"}],"fun_headline_variants":["Exactly one in k+1 prime powers is representable","Prime powers: one in k+1 are cx+dy representable","Frobenius problem: asymptotically one in k+1 prime powers","Prime powers up to cd-c-d: one in k+1 are representable","Asymptotic count: prime powers representable as cx+dy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Exactly one in k+1 prime powers is representable","Prime powers: one in k+1 are cx+dy representable","Frobenius problem: asymptotically one in k+1 prime powers","Prime powers up to cd-c-d: one in k+1 are representable","Asymptotic count: prime powers representable as cx+dy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000265,"raw_usage":{"total_tokens":1571,"prompt_tokens":870,"completion_tokens":701,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":486,"completion_tokens_details":{"reasoning_tokens":606}},"tokens_in":486,"tokens_out":701,"duration_ms":4844,"temperature":1.0,"reasoning_tokens":606,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T04:24:19.386432+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}