{"id":"038e7b3e-f55e-4aa1-98b9-9f3049e918ac","arxiv_id":"1908.06161","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For every m ≥ 2 there is a string of m consecutive primes in which every pair is symmetric, and the number of symmetric primes up to x is bounded by π(x)/(log x)^η (log log x)^O(1) with the conjectured exponent η.","lead":"This number theory paper proves two long-conjectured results about symmetric prime pairs: the conjectured upper bound on how many primes belong to such pairs, and the existence of arbitrarily long runs of consecutive primes in which every pair is symmetric. It improves a 1996 theorem and opens new questions about the graph of symmetric primes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1.1 relies on an unproved uniform Brun-type bound for R(x,d,m); if that bound hides an extra factor loss, the upper bound collapses, but the estimate is a standard three-prime sieve and plausibly correct.","rationale":"The reader's weakest assumption is exactly the uniform sieve bound for R(x,d,m), and my reading confirms this is the least secure input in the paper. However, a careful look at the sieve dimension shows the bound is the natural one for a three-linear-form problem: the range r ≤ x/(dm) has log(x/(dm)) ≥ (log x)/log log x, so X/(log X)^3 is essentially X/(log x)^3 times a power of log log x, and the singular series from primes dividing d, m, and m±1 is also (log log x)^O(1). Thus the asserted R bound is plausible and standard in spirit, though the paper does not supply the derivation. I found no internal inconsistency in the rest of the argument. In particular, Theorem 1.2 is sound: the divisibility step works because e = |h_i−h_j| divides both h_i−1 and h_j−1, hence divides g = ∏(b_l−1), so e divides both P_i−1 and P_j−1; conversely any common divisor of P_i−1 and P_j−1 divides their difference h_i−h_j. The admissibility of the tuples is correctly justified, and the use of BFTB is legitimate after proving gcd(g, b_1...b_k)=1. Therefore the unproved sieve bound is a genuine technical gap but not, on the evidence, a correctness failure; the verdict should remain ACCEPT.","tokens_in":5795,"tokens_out":20644,"duration_ms":213276,"concrete_test":"Derive R(x,d,m) from a standard linear sieve (e.g., the Fundamental Lemma or Iwaniec's linear sieve, Halberstam-Richert Chapter 2), tracking all local factors. For primes r ≤ X, X = x/(dm), with dmr+1 prime and at least one of dmr ± d + 1, dr(m±1)+1 prime, show the upper bound is C X/(log X)^3 ∏_p (1−ν_p/p)/(1−1/p)^3, and verify that the product is (log log x)^O(1) uniformly for dm ≤ x^{1−1/log log x}. If the replacement log X ≥ (log x)/log log x leaves only an extra (log log x)^C factor, the claimed bound holds and the Section 2 sum is valid; if any factor (log X)^{-3} is replaced by something smaller, the proof of Theorem 1.1 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single most load-bearing assumption is the assertion in Section 2 that R(x,d,m) ≤ x/(dm(log x)^3)(log log x)^O(1) uniformly for dm < x^{1−1/log log x}, justified only by 'Again by Brun's method'. This bound is not proved, and no precise citation for this exact configuration is given. It is exactly what converts the divisor-sum estimate ∑_{Ω(dm)<L} 1/(dm) ≪ (log x)^{2−η}(log log x)^O(1) into the claimed S(x) ≤ π(x)/(log x)^η(log log x)^O(1); losing any power of log x, or getting a factor depending on log(x/(dm)) in the wrong direction, would break the theorem. The concern is not that the bound is false: counting primes r ≤ X with dmr+1 and one shifted linear form prime is a three-linear-form sieve problem, so the natural main term is X/(log X)^3 times a singular series of size (log log x)^O(1), and in the stated range log X ≥ (log x)/log log x. The issue is that the paper leaves this load-bearing input unverified, so the central upper bound rests on an unstated technical lemma.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies symmetric pairs of odd primes, defined by gcd(p−1, q−1) = |p−q|. It proves two main results. Theorem 1.1 gives the upper bound S(x) ≤ π(x)/(log x)^η (log log x)^{O(1)} for all large x, where η = 1 − (1 + log log 2)/log 2 = 0.08607..., improving the 1996 Fletcher–Lindgren–Pomerance upper bound and confirming the conjecture from that paper. Theorem 1.2 states that for every integer m ≥ 2 there is a string of m consecutive primes, any two of which form a symmetric pair, which implies the infinitude of symmetric primes. The proof of Theorem 1.1 splits primes according to P^+(p−1) and Ω(p−1), uses two Brun-type sieve estimates and Hall–Tenenbaum divisor sums, and reduces the main term to an exponential divisor-sum bound. The proof of Theorem 1.2 combines sets with the property gcd(a,b)=|a−b|, the Maynard–Tao theorem, and a theorem of Banks–Freiberg–Turnage-Butterbaugh on consecutive primes in admissible tuples.","tokens_in":5905,"tokens_out":15903,"duration_ms":153154,"significance":"If the proofs are correct, this is a substantial advance: it resolves the 1996 conjecture on the exponent in the upper bound for symmetric primes and answers the long-standing question of whether infinitely many symmetric primes exist, in the strong form of arbitrarily long consecutive-prime strings. The architecture of the proof is sensible: the S1/S2/remaining split is natural, the divisor-sum calculation is clean and yields exactly the exponent 2−η, and the use of Hall–Tenenbaum estimates is well matched to the problem. The paper also contains new computations of S(x) up to the 10^8-th prime. The main caveat is that the central upper bound relies on a uniform Brun-type estimate for R(x,d,m) that is asserted but not proved or precisely cited; this point is load-bearing and needs to be made explicit for the proof to be fully rigorous.","major_comments":[{"comment":"The estimate for R(x,d,m) is load-bearing for Theorem 1.1. Please add a proof or a precise citation with the required uniformity.","section":"Section 2, estimate for R(x,d,m)"}],"minor_comments":[{"comment":"Theorem 3.4 is stated for k = k_m, but it is subsequently applied with k ≥ k_m. The intended argument is clear, since a subset of size k_m of an admissible tuple remains admissible and preserves the gcd property, but the text should either state the theorem for all k ≥ k_m or explicitly pass to a subset of size k_m before applying it.","section":"Section 3, application of Theorem 3.4"},{"comment":"The quantity E is defined as the reciprocal sum of all primes and prime powers less than x, while the inner sums restrict d and m to be less than x. The inequality ∑_{ω(d)=i} 1/d ≤ E^i/i! is valid, but it would help the reader to note that using the full E is an upper bound and that the restriction to d < x is not used in the estimate.","section":"Section 2, divisor-sum calculation"},{"comment":"The proof of Lemma 3.2 invokes a theorem of Maynard and 'Tao (unpublished)'; since part of this result is unpublished, the paper should provide a precise reference for the full statement used here, in addition to the citation to Maynard's paper.","section":"Section 3, Maynard–Tao attribution"},{"comment":"The sentence comparing the descent in S(p_n)/n with 'the main term in our upper bound' is informal; it would be clearer to state explicitly that the comparison is with (log p_n)^{−η}, as shown in the last column of Table 1.","section":"Section 4, Table 1"}],"recommendation":"major_revision","confidential_remarks":"I see no grounds for rejection: the central arguments are coherent and the only serious gap is the unproved uniform sieve estimate for R(x,d,m) in Section 2, which is standard in spirit but must be made explicit. The mismatch in the application of Theorem 3.4 is a minor fix. The paper is well suited to the journal's readership."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Paul, this one is worth your time. Banks, Pollack, and Pomerance resolve both open problems from the 1996 Fletcher–Lindgren–Pomerance paper: they prove the conjectured power-of-log upper bound with the sharp exponent eta = 0.08607, and they show that symmetric primes occur in arbitrarily long runs of consecutive primes. That is a real advance, and the exposition is unusually clear for a paper this dense.\n\nThe proof of Theorem 1.1 is a clean three-case sieve argument. The decomposition into S1, S2, and the remaining middle case is standard, and the divisor-sum computation via Hall–Tenenbaum bounds is neat. Theorem 1.2 is the more surprising: combining Heath-Brown's gcd sets with Maynard–Tao and the Banks–Freiberg–Turnage-Butterbaugh theorem on consecutive primes is a genuinely new construction, and the admissibility check is clever.\n\nThe main soft spot is exactly what the stress-test flagged: the uniform Brun-type bound for R(x,d,m) in Section 2 is asserted as \"Again by Brun's method\" with no proof and no precise citation. This bound is load-bearing, because it supplies the extra (log x) factor that converts the divisor sum into the claimed upper bound. If it failed in the required uniformity, Theorem 1.1 would collapse. That said, the estimate is a standard three-linear-forms sieve problem, the range dm < x^{1-1/log log x} leaves log X at least (log x)/log log x, and the singular series is no larger than (log log x)^{O(1)}. I am fairly confident the bound is true and can be proved by adapting Halberstam–Richert, but the paper as written leaves it as an unstated lemma.\n\nThe reliance on Maynard–Tao and BFTB is appropriate; these are published external results, and the self-citation to [1] is legitimate since it is the exact tool needed. The computations in Section 4 are a nice sanity check, though they do not strongly discriminate between the old and new exponents.\n\nBottom line: the central results are very likely correct, and the proof is structurally sound modulo that one sieve estimate. This deserves a serious referee, who should ask the authors to supply a proof or reference for the Brun bound. I would accept it for peer review with that request.","headline":"A short, elegant paper that settles the two main open problems on symmetric primes; the only serious soft spot is a load-bearing sieve estimate that is asserted but not proved.","tokens_in":6594,"tokens_out":2758,"would_cite":true,"duration_ms":25893,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N05","11N36","11N37","11A41"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves the count of symmetric primes up to x is at most π(x)/(log x)^η with η≈0.086, and that arbitrarily long consecutive-prime strings are pairwise symmetric.","keywords":["symmetric primes","asymmetric primes","gcd condition","prime tuples","sieve methods","divisor sums","consecutive primes","multiplication table constant"],"falsifier":"The most direct falsifier is a count: any sequence $x_j\\to\\infty$ with $S(x_j)/\\pi(x_j)$ failing to decay at least as fast as $(\\log x_j)^{-\\eta}(\\log\\log x_j)^{O(1)}$ would disprove Theorem 1.1. Since finite data cannot settle this, the practical check is the proof's only unproved ingredient: verify the asserted uniform estimate $R(x,d,m)\\le x/(dm(\\log x)^3)(\\log\\log x)^{O(1)}$ for all $d,m$ with $dm<x^{1-1/\\log\\log x}$ and $\\Omega(dm)<L$; an infinite family violating that estimate would remove the support of the upper bound, though not necessarily the theorem itself.","tokens_in":5441,"feed_emoji":"🔢","tokens_out":11366,"duration_ms":101537,"temperature":0.7,"pith_summary":"Two old problems about symmetric primes — pairs of odd primes $\\{p,q\\}$ with $\\gcd(p-1,q-1)=|p-q|$ — are settled here. First, the number $S(x)$ of symmetric primes up to $x$ is at most $\\pi(x)/(\\log x)^{\\eta}\\,(\\log\\log x)^{O(1)}$, where $\\eta=1-(1+\\log\\log 2)/\\log 2\\approx 0.08607$; this proves the upper-bound conjecture from the 1996 paper that introduced the term. Second, for every $m\\ge 2$ there is a block of $m$ consecutive primes in which every two primes form a symmetric pair, so in particular infinitely many symmetric primes exist. The interest is that symmetric primes are sparse but not too sparse, with the same exponent $\\eta$ that governs the multiplication-table problem.","feed_headline":"A 1996 conjecture on symmetric primes is proved","feed_subtitle":"The primes are sparse, yet for every m there are m consecutive primes that are pairwise symmetric.","key_machinery":"The argument for Theorem 1.1 splits primes $p\\le x$ according to the largest prime factor $r$ of $p-1$ and the number $\\Omega(a)$ of prime factors of $a=(p-1)/r$. If $p$ is symmetric, some divisor $d$ of $a$ has one of $p\\pm d$, $p\\pm dr$ prime; this reduces the count to a uniform Brun-sieve estimate $R(x,d,m)$ for primes $r$ with $dmr+1$ prime and a shifted value also prime. Summing that estimate over pairs $d,m$ with few prime factors, using the divisor-sum bounds in [7], produces the extra $(\\log x)^{-\\eta}$ factor and the constant $\\eta$. For Theorem 1.2, the machinery is a finite set $A$ with pairwise gcd equal to difference (an example for size four is $\\{6,8,9,12\\}$): if $an+1$ and $bn+1$ are prime for $a,b\\in A$, then the two primes are symmetric. The bounded-gaps theorem guarantees infinitely many $n$ for which the $k$ forms $an+1$ contain $m$ primes, and a separate theorem on consecutive primes in admissible tuples converts those $m$ primes into a string of consecutive primes.","core_discovery":"On the paper's own terms, the central discovery is a two-part refinement of the 1996 results. Theorem 1.1 gives $S(x)\\le \\pi(x)/(\\log x)^{\\eta}(\\log\\log x)^{O(1)}$ for all large $x$, with $\\eta=1-(1+\\log\\log 2)/\\log 2$, improving the old upper bound whose exponent was only $0.027$ and matching the exponent conjectured earlier. Theorem 1.2 shows that for each $m\\ge 2$ one can find $m$ consecutive primes $p_1<\\dots<p_m$ with $\\gcd(p_i-1,p_j-1)=|p_i-p_j|$ for all $i\\ne j$, so every pair is symmetric. The proof of Theorem 1.2 constructs finite sets $A$ of integers with $\\gcd(a,b)=|a-b|$ for distinct $a,b$ (for size four, $\\{6,8,9,12\\}$), turns the linear forms $an+1$ into primes via a bounded-gaps theorem for admissible tuples, and then applies a result on consecutive primes in admissible tuples to make the resulting primes consecutive. This answers the open infinitude problem in the strong form that symmetric primes occur in arbitrarily long consecutive clusters.","pith_inferences":["The paper leaves implicit that the same $\\eta$ appearing in the multiplication-table problem is not a coincidence: both estimates are controlled by divisor sums over integers with restricted numbers of prime factors, so a sharp form of one bound may transfer to the other.","A wide gap remains between the proved upper-bound exponent $0.086$ and the lower-bound exponent $49$ mentioned in the paper; improving the uniform sieve estimate or the divisor-sum step is the natural route to narrowing it.","The sets $A$ with $\\gcd(a,b)=|a-b|$ are of independent combinatorial interest; studying the minimal size of such sets and the least $n$ for which the $an+1$ are prime would give computable lower bounds for the first $m$-consecutive symmetric cluster.","The graph questions in Section 5 suggest a testable extension: computing the connected component containing the prime 3 far beyond the paper's table could indicate whether symmetric primes form one giant component or many, and whether every connected component is finite."],"forward_implications":["If Theorem 1.1 is right, symmetric primes have natural density zero among primes: $S(x)/\\pi(x)\\to 0$ as $x\\to\\infty$, albeit slowly.","The upper-bound exponent $\\eta$ matches the exponent conjectured in the 1996 paper, and the authors' heuristic predicts the matching lower bound $S(x)=\\pi(x)/(\\log x)^{\\eta+o(1)}$.","Theorem 1.2 implies infinitely many symmetric primes, and in fact for every $m$ the graph on odd primes whose edges are symmetric pairs contains a $K_m$ with its vertices consecutive primes.","Because any symmetric pair with $p<q$ must satisfy $q<2p$, the graph contains no infinite complete subgraph even though it contains arbitrarily large finite ones."],"supporting_citations":[{"why":"defines symmetric pairs, proves the old upper bound with exponent 0.027, states the conjecture Theorem 1.1 proves, and poses the infinitude problem Theorem 1.2 answers.","marker":"[4]"},{"why":"gives the smooth-number estimate used to dispose of primes p with all prime factors of p−1 small.","marker":"[2]"},{"why":"supplies the Brun sieve bound for primes r with ar+1 prime, used for the easy part of S2(x).","marker":"[6]"},{"why":"provides the divisor-sum theorems that turn sums of 1/a over a with restricted Ω(a) into (log x)^{1−η}.","marker":"[7]"},{"why":"gives the Heath-Brown result on the divisor function at consecutive integers from which Lemma 3.1's sets A_k derive.","marker":"[8]"},{"why":"introduces the construction behind Lemma 3.1, sets with pairwise gcd equal to difference, via Spiro's dissertation.","marker":"[14]"},{"why":"supplies the bounded-gaps theorem for admissible tuples used to turn the forms an+1 into many primes.","marker":"[11]"},{"why":"provides the theorem on consecutive primes in admissible tuples that upgrades the m primes to consecutive primes.","marker":"[1]"}],"fun_headline_variants":["Optimal bound and arbitrary runs for symmetric primes","Symmetric primes found in arbitrarily long consecutive runs","Conjectured symmetric prime bound proved optimal","Longest symmetric prime runs: arbitrarily long, proved","Symmetric primes: optimal counting and infinite clusters"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the uniform Brun-sieve estimate $R(x,d,m)\\le x/(dm(\\log x)^3)(\\log\\log x)^{O(1)}$, asserted without proof for all $d,m$ in the required range; if that estimate fails in the needed uniformity, the proof of Theorem 1.1 does not go through.","fun_headline_variants_meta":{"raw":{"variants":["Optimal bound and arbitrary runs for symmetric primes","Symmetric primes found in arbitrarily long consecutive runs","Conjectured symmetric prime bound proved optimal","Longest symmetric prime runs: arbitrarily long, proved","Symmetric primes: optimal counting and infinite clusters"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000479,"raw_usage":{"total_tokens":2344,"prompt_tokens":892,"completion_tokens":1452,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":508,"completion_tokens_details":{"reasoning_tokens":1381}},"tokens_in":508,"tokens_out":1452,"duration_ms":8888,"temperature":1.0,"reasoning_tokens":1381,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:55:11.376552+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The most direct falsifier is a count: any sequence $x_j\\to\\infty$ with $S(x_j)/\\pi(x_j)$ failing to decay at least as fast as $(\\log x_j)^{-\\eta}(\\log\\log x_j)^{O(1)}$ would disprove Theorem 1.1. Since finite data cannot settle this, the practical check is the proof's only unproved ingredient: verify the asserted uniform estimate $R(x,d,m)\\le x/(dm(\\log x)^3)(\\log\\log x)^{O(1)}$ for all $d,m$ with $dm<x^{1-1/\\log\\log x}$ and $\\Omega(dm)<L$; an infinite family violating that estimate would remove the support of the upper bound, though not necessarily the theorem itself.","supporting_citations":[{"cited_title":"Fletcher, W","cited_arxiv_id":null,"evidence_quote":"defines symmetric pairs, proves the old upper bound with exponent 0.027, states the conjecture Theorem 1.1 proves, and poses the infinitude problem Theorem 1.2 answers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the smooth-number estimate used to dispose of primes p with all prime factors of p−1 small."},{"cited_title":"Halberstam and H.-E","cited_arxiv_id":null,"evidence_quote":"supplies the Brun sieve bound for primes r with ar+1 prime, used for the easy part of S2(x)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the divisor-sum theorems that turn sums of 1/a over a with restricted Ω(a) into (log x)^{1−η}."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the Heath-Brown result on the divisor function at consecutive integers from which Lemma 3.1's sets A_k derive."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces the construction behind Lemma 3.1, sets with pairwise gcd equal to difference, via Spiro's dissertation."},{"cited_title":"Maynard, Dense clusters of primes in subsets, Compositio Math","cited_arxiv_id":null,"evidence_quote":"supplies the bounded-gaps theorem for admissible tuples used to turn the forms an+1 into many primes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the theorem on consecutive primes in admissible tuples that upgrades the m primes to consecutive primes."}],"review_version":1}