{"id":"e768a05b-aa5a-4e60-8e35-f2e7bd579062","arxiv_id":"2411.17599","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every non-integer c>2, the primes contain infinitely many pairs (p, p+floor(m^c)), with the expected asymptotic density 1.","lead":"For any non-integer exponent c>2, the authors prove that prime pairs with difference equal to the integer part of m^c occur with the expected frequency. This gives the first proof that the primes contain infinitely many pairs separated by such fractional-power differences.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.5 drops the log^4 factor from Vinogradov's bound, so Proposition 3.2 only yields O(log^{-A+4/k}N); the gap is real but repairable by enlarging b.","rationale":"I read the paper in good faith. The Fourier strategy is credible: the main term, the Parseval estimates, Lemma 3.7, and the van der Corput part of Section 3.2 appear sound. The reader's concern is exactly where the proof's quantitative control of Σ1 lives. I independently verified the log^4 loss: Vinogradov's log factor multiplies the whole bracket, so after q > log^{b/2}N it cannot be discarded. The chosen b only provides the +1 needed to cancel the Parseval log N; it does not provide the extra 4/k needed to absorb log^4. This is an internal gap as written, not merely a disagreement with consensus, and it directly affects the central claim. However, because b is an auxiliary parameter unrelated to M, enlarging it by O(k) restores the bound; nothing in the argument suggests the theorem is false. I therefore keep the reader's CONDITIONAL verdict. I do not treat the evident notational slips (complex arguments in Lemma 3.9, the symmetry factor in Eq. (3.1)) as load-bearing, since they are presentation errors harmless to the estimates.","tokens_in":17356,"tokens_out":25193,"duration_ms":218066,"concrete_test":"Recompute the final display of Lemma 3.5 with the log^4 factor retained, giving sup |hat Lambda(xi)| = O(N log^4 N / log^{b/4}N) = O(N/log^{b/4-4}N). Then re-run the Hölder estimate in Proposition 3.2 symbolically: for b = 4kA + 2Bc + 4k the final exponent is -A + 4/k, not -A. As a repair check, verify that replacing b by 4kA + 2Bc + 20k makes the final exponent ≤ -A for all A ≥ 1; if so, the theorem holds with a modified auxiliary parameter and no change to M.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 rests on Proposition 3.2, whose exponent is driven by Lemma 3.5. In Lemma 3.5, Vinogradov's bound (Theorem 2.6) gives |hat Lambda(xi)| ≤ C(N/q^{1/2} + N^{4/5} + (qN)^{1/2}) log^4 N. With q > log^{b/2}N this is at most C N log^4 N / log^{b/4}N, that is, O(N/log^{b/4-4}N), not O(N/log^{b/4}N). The proof as printed omits the log^4 factor from the first two terms and concludes O(N/log^{b/4}N). With the chosen b = 4kA + 2Bc + 4k, the supremum term in Hölder's inequality contributes exponent -(b/4 - 4)/k = -(A + Bc/(2k) + 1 - 4/k). Combined with Lemma 3.4's O(log N) and Lemma 3.7's O(log^{Bc/(2k)}N), Proposition 3.2 yields O(log^{-A+4/k}N), which is weaker than the claimed O(log^{-A}N) because 4/k > 0. This is the load-bearing step: if it fails, Σ1 is not controlled at the required strength. The flaw is localized and appears repairable by taking b larger, so it does not invalidate the theorem if the auxiliary parameter is adjusted.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Cesàro average E_{m≤M} E_{n≤N} Λ(n)Λ(n+⌊m^c⌋) for non-integer c>2 and M=N^{1/c}/log^B N. Theorem 1.1 claims this average equals 1+O(log^{-A}N) for any A≥1 when B=(A+2)/c, and Corollary 1.2 concludes that the primes contain infinitely many pairs whose difference belongs to the Piatetski-Shapiro sequence {⌊m^c⌋}. The proof rewrites the average via the discrete Fourier transform on Z/NZ (Proposition 2.10), decomposes the resulting character sum into a main term, a 'close to trivial' part Σ1, and a 'far' part Σ2, and estimates Σ1 using Parseval, Vinogradov's bound for the Fourier transform of Λ, and Poulias's Diophantine inequality, and Σ2 using the Erdős-Turán inequality and van der Corput estimates.","tokens_in":17655,"tokens_out":21112,"duration_ms":164550,"significance":"If the proof is completed, the result is a natural complement to the Tao–Ziegler theorems on polynomial patterns in the primes: it gives the first asymptotic for prime pairs with difference in a non-integer power sequence, and the frequency matches the naive independence heuristic. The paper is clearly structured, gives a dependency graph, and states precise conjectures for generalizations. The main term is benchmarked against the Prime Number Theorem, and the argument is built from external standard estimates rather than assuming the conclusion; there is no evidence of circularity or fitted constants. The identified gaps are localized and appear repairable, so the work has the potential to be a publishable contribution after revision.","major_comments":[{"comment":"The proof of Lemma 3.5 applies Theorem 2.6 to obtain |Λhat(ξ)| ≤ C(N/√q + N^{4/5} + √(qN)) log^4 N. Since q > log^{b/2}N, the first term is at most N log^4 N / log^{b/4}N, that is, N/log^{b/4-4}N, not N/log^{b/4}N as written. The displayed estimate after applying Vinogradov therefore does not yield the stated bound O(N/log^{b/4}N). With the paper's choice b=4kA+2Bc+4k, the Hölder step in Proposition 3.2 produces O(log^{-A+4/k}N), which is weaker than the claimed O(log^{-A}N). This is load-bearing because Lemma 3.5 controls the entire contribution of Σ1. The flaw is localized and repairable by enlarging b (for instance b=4kA+2Bc+4k+16), but the proof as written does not deliver the theorem.","section":"Section 3.1, Lemma 3.5"},{"comment":"The displayed identity (3.1) is false as written. The left-hand side sums over ξ=0,...,N-1, while the right-hand side is the main term plus Σ1 over 1≤ξ≤N/log^b N plus Σ2 over N/log^b N≤ξ≤N/2, omitting the range N/2<ξ≤N-1. The sentence 'Since |Λhat(ξ)|=|Λhat(-ξ)|, we can reduce to ξ≤N/2' is not an equality; the omitted terms are the conjugates of the corresponding half-range terms and must be bounded separately or incorporated via 2Re. The final paragraph of Lemma 3.15 invokes symmetry, but Eq. (3.1) itself is incorrect. The decomposition should be rewritten with the symmetric contribution made explicit.","section":"Section 3, Eq. (3.1)"},{"comment":"The definitions of µ1, µ2 and c1, c2 use cos(-2πi ξm^c/N) and sin(-2πi ξm^c/N). With the imaginary unit inside, these are hyperbolic functions, not bounded by 1, and the claim that µ1 and µ2 are probability measures is false. The product-to-sum identities in Lemma 3.10 show that the intended arguments are the real quantities cos(2πξm^c/N) and sin(2πξm^c/N). Without correcting these definitions, Lemma 3.10 and hence Proposition 3.14 (and therefore Proposition 3.3) are not established. Additionally, the assertion |c1|,|c2|≤1/2 in Lemma 3.10 is not proved; the proof uses 1/(1+c1)≤2, so a uniform lower bound for 1+c1 is needed.","section":"Section 3.2, Lemmas 3.9 and 3.10"}],"minor_comments":[{"comment":"The symbols r and ξ are both used for the frequency, and k is re-used for ⌊log^m N⌋ in conflict with the global k in Theorem 2.8; this makes the proof harder to follow.","section":"Section 3.1, Lemma 3.6"},{"comment":"The phrase 'noting that j = 2 j+1 ≈ M' is garbled; it should say that the largest dyadic term occurs at j ≈ log_2 M.","section":"Section 3.2, Proposition 3.14"},{"comment":"The text says 'apply Fourier inversion to each of the indicator functions', but the objects being transformed are the von Mangoldt function, not indicators; this is a wording issue.","section":"Section 2.4, Proposition 2.10"},{"comment":"The proof should state explicitly that N is taken large enough so that kN/log^{Bc}N < N, since this is needed for the geometric sum evaluation to reduce to the R=0 case.","section":"Section 3.1, Lemma 3.7"}],"recommendation":"major_revision","confidential_remarks":"The errors identified are localized and the overall strategy is sound in outline, so I recommend major revision rather than rejection. I saw no problematic citation patterns, no circularity, and no fitted constants in the argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper proves a new asymptotic for prime pairs with Piatetski-Shapiro differences and gets the infinitude as a corollary. The Fourier strategy is sound, and I believe the theorem is true. But Lemma 3.5 drops a log^4 factor, and as written Proposition 3.2 gives O(log^{-A+4/k}), not O(log^{-A}). That is a real gap, though localized and easily repaired.\n\nWhat's actually new: Tao-Ziegler covers integer powers; this paper handles non-integer c>2 and gives the first infinitude result for prime pairs whose difference is floor(m^c). The decomposition into close/far characters, use of Poulias's Diophantine bound, and the Erdős-Turán plus van der Corput treatment of the far range are sensible and mostly standard. The paper is honest about what it does and does not do; the conjecture section is clearly labeled.\n\nSoft spots: the Lemma 3.5 issue is the main one. Vinogradov's bound has log^4 N on the whole bracket. With q > log^{b/2}, the first term contributes N/log^{b/4-4}, not N/log^{b/4}. The proof absorbs the log^4 only into the third term. Then the Hölder step in Proposition 3.2 has exponent -(b/4-4)/k instead of -b/4/k, and the final bound is O(log^{-A+4/k}). Since A is arbitrary, a modest enlargement of b (add 16) repairs it; the theorem is likely fine. Eq. (3.1) also seems to miss the conjugate half of Σ2, which is a factor-of-two constant issue. The trig notation in Lemma 3.9 writes cos(-2πi ξ m^c/N) with an i inside cos; that should be cos(2πξ m^c/N). Minor. Also k should be the ceiling, not floor, in the definition before Theorem 3.1 to satisfy Poulias's strict inequality. The references look appropriate; no circularity.\n\nWho this is for: analytic number theorists interested in additive patterns in primes. I'd send it to a serious referee. The referee should verify the log power fixes and the notation cleanup, but the core idea is solid.","headline":"A genuinely new result on prime pairs with fractional-power differences, but the proof currently has a repairable log-power gap in Lemma 3.5 that should be fixed before publication.","tokens_in":18181,"tokens_out":7947,"would_cite":false,"duration_ms":93928,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N05","11L07","11L20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For non-integer c>2, the primes contain infinitely many pairs separated by floor(m^c).","keywords":["prime differences","Piatetski-Shapiro sequences","fractional powers","von Mangoldt function","Fourier analysis on Z/NZ","exponential sums","Diophantine inequalities","multiple correlations in primes"],"falsifier":"Re-derive Proposition 3.2 while keeping the $\\log^4 N$ factor from Theorem 2.6 visible: Lemma 3.5's bound becomes $O(N\\log^4 N/\\log^{b/2}N)$, and with $b=4kA+2Bc+4k$ the Hölder step yields only $O(\\log^{-(A+(Bc-4)/k)}N)$. If this computation is carried through, the claimed $O(\\log^{-A}N)$ conclusion is not established, and the proof would need a sharper estimate or a different split.","tokens_in":17164,"feed_emoji":"🔢","tokens_out":10251,"duration_ms":86694,"temperature":0.7,"pith_summary":"This paper claims a new asymptotic for the frequency of prime pairs whose difference is a fractional power. For any non-integer c>2, the Cesàro average of $\\Lambda(n)\\Lambda(n+\\lfloor m^c\\rfloor)$ over n≤N and m≤M is shown to be $1+O(\\log^{-A}N)$, with $M=N^{1/c}/\\log^B N$ and $B=(A+2)/c$. If correct, this is the first proof that the primes contain infinitely many pairs (p,m) with both p and p+⌊m^c⌋ prime, at the expected density. The paper leaves c in (1,2) open and treats c>2 as a technical restriction, conjecturing the result should hold for all c>0.","feed_headline":"Primes have infinitely many fractional-power differences","feed_subtitle":"A new proof shows prime pairs differ by floor(m^c) at the expected frequency.","key_machinery":"The proof uses Fourier analysis on $\\mathbb{Z}/N\\mathbb{Z}$, rewriting the double average as a character sum of $|\\widehat{\\Lambda}(\\xi)|^2$ against an exponential sum in $\\lfloor m^c\\rfloor$. The trivial character contributes the main term, and the nontrivial characters are split into a 'close' range, $\\xi\\le N/\\log^b N$, and a 'far' range, $\\xi>N/\\log^b N$. For the close range, Hölder's inequality separates the Fourier mass of $\\Lambda$ from a 2k-th moment of the exponential sum, and that moment is bounded by counting solutions to the Diophantine inequality $|\\sum m_i^c - \\sum m_j^c|\\le k$. For the far range, the maximum of the exponential sum is factored out, the floor function is replaced by $m^c$ through a partition of the circle, a discrepancy inequality controls the replacement, and classical exponential-sum bounds for smooth phases finish the estimate.","core_discovery":"The central claim is Theorem 1.1: for any non-integer c>2 and any A≥1, with B=(A+2)/c and M=$N^{{1/c}}$/\\log^B N, the average $E_{m\\le M}E_{n\\le N} \\Lambda(n)\\Lambda(n+\\lfloor m^c\\rfloor)$ equals $1+O(\\log^{-A}N)$. The main term 1 is exactly what one would get if $\\Lambda(n)$ and $\\Lambda(n+\\lfloor m^c\\rfloor)$ behaved independently, and the error shrinks faster than any fixed power of $1/\\log N$. A direct consequence is Corollary 1.2: infinitely many pairs (p,m) with p and p+⌊m^c⌋ both prime, and in fact about $N M \\log^{-2}N$ such pairs with p≤N and m≤M. This is the fractional-power analogue of the known polynomial-pattern asymptotics for the primes.","pith_inferences":["Editorial extension: The same Fourier-decoupling scheme should transfer to multiple fractional-power differences, for instance differences of the form $\\lfloor m^{c_1}\\rfloor+\\lfloor m^{c_2}\\rfloor$, if a fractional analogue of the quoted Diophantine bound exists; this is the direction the paper's Conjecture 4.2 points toward.","Editorial extension: Numerical evaluation of the double average for c=2.5 or c=3.5 at moderate N would expose whether the logarithmic error is real and how large the implied constant is; the paper reports no such checks.","Editorial extension: If the close-character estimate loses a few powers of log, a slightly weaker theorem with A replaced by $A-(Bc-4)/k$ may still hold, because the Hölder decomposition absorbs logarithmic losses without changing the structure."],"forward_implications":["For every non-integer c>2, the primes contain infinitely many pairs (p,m) with p and p+⌊m^c⌋ both prime.","The number of such pairs with p≤N and m≤M is approximately $N M \\log^{-2}N$, matching the heuristic independence count.","The double average is $1+o(1)$, the same leading constant as for polynomial patterns, so fractional powers introduce no extra bias at this order.","For any fixed A, taking $M=N^{1/c}/\\log^{(A+2)/c}N$ makes the error $O(\\log^{-A}N)$; increasing A improves the error only by shortening the m-range."],"supporting_citations":[{"why":"Supplies the asymptotic for polynomial patterns in the primes that this paper's fractional-power average is designed to extend.","marker":"[TZ18]"},{"why":"Gives the bound on solutions of a Diophantine inequality for fractional powers used to estimate the 2k-th moment of the close-character exponential sum.","marker":"[Pou21]"},{"why":"Provides the quoted Fourier estimate for the von Mangoldt function that bounds close characters in Lemma 3.5.","marker":"[Dav67]"},{"why":"Supplies the Prime Number Theorem estimate used for the main term and for the Fourier transform near the trivial character.","marker":"[MV06]"},{"why":"Provides one half of the discrepancy inequality used to control the approximation of the floor-function exponential sums.","marker":"[ET48a]"},{"why":"Provides the second half of the discrepancy inequality, cited as Theorem 3 in Lemma 3.12.","marker":"[ET48b]"},{"why":"Provides the exponential-sum estimate for smooth phases used to bound the far characters.","marker":"[GK91]"}],"fun_headline_variants":["Prime pairs have infinite fractional-power differences","Infinitely many primes differ by floor(m^c) for c>2","Fractional power gaps between primes are infinite","New proof: prime gaps match fractional powers infinitely often","Primes show infinite pairs with gaps floor(m^c)"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the uniform bound on the Fourier transform of the von Mangoldt function over the close characters holds with the full logarithmic savings stated in Lemma 3.5; if the extra $\\log^4 N$ factor from the cited Fourier estimate is not absorbed, the final $O(\\log^{-A}N)$ error term does not follow as written.","fun_headline_variants_meta":{"raw":{"variants":["Prime pairs have infinite fractional-power differences","Infinitely many primes differ by floor(m^c) for c>2","Fractional power gaps between primes are infinite","New proof: prime gaps match fractional powers infinitely often","Primes show infinite pairs with gaps floor(m^c)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001077,"raw_usage":{"total_tokens":4476,"prompt_tokens":880,"completion_tokens":3596,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":496,"completion_tokens_details":{"reasoning_tokens":3519}},"tokens_in":496,"tokens_out":3596,"duration_ms":26459,"temperature":1.0,"reasoning_tokens":3519,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:57:14.539599+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-derive Proposition 3.2 while keeping the $\\log^4 N$ factor from Theorem 2.6 visible: Lemma 3.5's bound becomes $O(N\\log^4 N/\\log^{b/2}N)$, and with $b=4kA+2Bc+4k$ the Hölder step yields only $O(\\log^{-(A+(Bc-4)/k)}N)$. If this computation is carried through, the claimed $O(\\log^{-A}N)$ conclusion is not established, and the proof would need a sharper estimate or a different split.","supporting_citations":[],"review_version":1}