{"id":"170d1618-39b1-4113-b76f-c9e21c72001c","arxiv_id":"2509.09745","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"A recurrence-based sequence n^2 - n - 1 divided by a gcd with a recursive helper is observed to yield only 1s and primes for the first 10,000 terms, but the prime-only claim is unproven.","lead":"A newly defined sequence divides a quadratic formula by a greatest common divisor and appears to output only 1s and primes, checked for the first 10,000 terms. The paper claims this is more efficient than a known prime-generating sequence, but the key prime-only property is only conjectured, not proven.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Key gcd identity is true and implies the 1-or-prime claim; the paper's proof gap is fillable, but the efficiency claim remains unquantified.","rationale":"The reader correctly identified the unproved gcd simplification as the key assumption and noted that the central claim is not demonstrated in the paper. But the assumption is true: it follows from Theorem 2's closed form for b(n), and the resulting divisibility argument proves the 1-or-prime property without needing the partial-cancellation case to be checked computationally. Thus the mathematical core of the paper survives scrutiny. Nevertheless, the paper as written remains unacceptable as a proof: the identity is asserted, not proved; the partial-cancellation conclusion is stated as if immediate from computation; and the efficiency comparison with Rowland's sequence is not defined, so the advertised practical/novelty claim is vacuous. A revision that inserts the missing proof and quantifies the efficiency comparison would make the paper acceptable; hence I would adjust the reader's REJECT to CONDITIONAL rather than leave it unchanged.","tokens_in":4240,"tokens_out":30406,"duration_ms":310310,"concrete_test":"Derive the congruence above from b(n)=(n+2)!(n+1)/2. Then, for n=3..1000, compute both gcd(n^2-n-1,b(n-3)+n b(n-4)) and gcd(n^2-n-1,(n-1)!) and check they agree; also factor x and confirm a(n) equals the unique prime factor of x exceeding n-1, if any. A single mismatch would falsify the central claim; if all match, the unproved identity and the 1-or-prime claim are confirmed.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The load-bearing spot is Section 1's unproved assertion that b(n-3)+n b(n-4) can be replaced by (n-1)! inside the gcd. The paper gives no proof, and the entire Section 2 classification depends on it. However, this assertion is actually correct and can be proved from the paper's own Theorem 2. Writing !m for the left factorial sum_{k<m} k!, Theorem 2 gives b(n)=(n+2)!(n+1)/2. Then b(n-3)+n b(n-4) = [x * !(n-3) + (n-1)(n-3)!]/2, where x=n^2-n-1. Since x is odd and gcd(x,n-2)=1, reducing modulo x yields 2(n-2)(b(n-3)+n b(n-4)) ≡ (n-1)! (mod x), so gcd(x,b(n-3)+n b(n-4)) = gcd(x,(n-1)!). Hence a(n)=x/gcd(x,(n-1)!). The denominator removes exactly the prime factors of x that are ≤ n-1. Because x < n^2, x cannot have two prime factors both ≥ n; therefore a(n) is either 1 or a single prime. Thus the headline mathematical claim is sound. The real defect is that the manuscript does not supply this proof, and the abstract's claim that the sequence is 'more efficient' than Rowland's is never defined or quantified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines an integer sequence a(n) for n≥3 by a(n) = (n^2 − n − 1)/gcd(n^2 − n − 1, b(n−3) + n b(n−4)), where b(n) = (n+2)(b(n−1) − b(n−2)) with b(−1)=0, b(0)=1. The abstract asserts that a(n) contains only 1's and primes, that it has a finite continued fraction representation, and that it is 'more efficient' than Rowland's prime-generating recurrence. Section 2, however, labels the 1-or-prime property as Conjecture 1, supported only by a 10,000-term computational check. The paper also presents two continued fraction identities (Theorems 1 and 2) and a family of generalized sequences a_k(n).","tokens_in":4642,"tokens_out":19020,"duration_ms":182230,"significance":"If the 1-or-prime property is established, this construction gives a compact and notably dense prime-producing sequence with a simple quadratic numerator, and the conjectured symmetry a(p−n+1)=p is interesting and easily testable. The manuscript verifies the first 10,000 terms and links the sequence to OEIS entries (A356247 and related), which is useful. However, as it stands the paper does not prove its central claim, and the continued-fraction theorems contain serious gaps and apparent errors. The value of the submission is currently that of a well-documented conjecture rather than a proved theorem.","major_comments":[{"comment":"The abstract states as a finding that a(n) contains only 1's and primes, but §2 explicitly formulates this as Conjecture 1 and offers only a 10,000-term computation. Since this is the central claim of the paper, the abstract must be corrected to describe the statement as conjectural, or a proof must be supplied.","section":"Abstract and §2, Conjecture 1"},{"comment":"The sentence 'It is immediate to observe that the combination b(n−3)+n b(n−4) can be replaced by (n−1)! in the greatest common divisor without altering the result' is the load-bearing simplification behind the §2 case analysis. No proof or reference is given. This gap must be closed. A proof can be supplied from the closed form for b(n) via the congruence 2(n−2)(b(n−3)+n b(n−4)) ≡ (n−1)! (mod n^2−n−1), but the manuscript does not contain it.","section":"§1, gcd replacement"},{"comment":"The proofs are not rigorous and the statements contain apparent errors. In Theorem 1's proof, Eq. (4) is algebraically wrong: for n=4 the recurrence gives a_1=3a_3−8a_4, not 3a_3−14a_4; the correct coefficient appears to be −n(n−2). Theorem 2 as stated fails for small n: for n=3 the left-hand side is 2m/(3m−6), while the right-hand side simplifies to 2(m−1)/(m−2). These identities need to be restated precisely and proved before the finite-continued-fraction claim in the abstract can be accepted.","section":"§3, Theorems 1 and 2"},{"comment":"The assertion that the sequence is 'more efficient for generating prime numbers than the Rowland sequence' is never defined or quantified. Specify the metric (e.g., proportion of non-1 terms that are prime, or density of primes up to N) and provide a numerical comparison. Without this, the claim is not falsifiable.","section":"§1 and Abstract, efficiency claim"}],"minor_comments":[{"comment":"The text says 'Where b(n)=...' should be 'where b(n)=...'.","section":"§1"},{"comment":"The three cases listed do not explicitly exclude the situation gcd(x,y)=1 with x composite. The text should note that Conjecture 1 asserts that this case does not occur.","section":"§2"},{"comment":"The sentence 'For 1≤k≤5, all terms in the sequence a_k(n)...' refers to the linear family in Table 2, not the quadratic family in Table 1. State this explicitly, since the k=3 row of Table 1 contains the composite value 9 and otherwise the claim appears contradictory.","section":"§4, after Table 2"},{"comment":"The parameter m in Theorems 1 and 2 is not defined; specify that it is a positive integer and the range of n for which the identities are intended.","section":"§3"},{"comment":"When listing the first sequence terms, state clearly that the indexing starts at n=3.","section":"§1"}],"recommendation":"major_revision","confidential_remarks":"This is a preliminary note in math.GM. The central 1-or-prime claim is likely true and fixable, but the manuscript as written does not prove it and contains concrete errors in the continued-fraction section. The author should be encouraged to supply the missing gcd proof, correct Theorems 1 and 2, and quantify the efficiency claim. The minimal reference list and lack of engagement with prior work (Rowland, Cloitre) should also be addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing you should know: the paper's headline observation is real, but the paper doesn't prove it. The author states that b(n-3)+n b(n-4) can be replaced by (n-1)! in the gcd as an \"immediate observation,\" gives no proof, and then builds the whole classification on it. I checked the claim myself. From Theorem 2 in the paper, b(n) = (n+2)!(n+1)/2, and a short modular argument gives gcd(x, b(n-3)+n b(n-4)) = gcd(x, (n-1)!) with x = n^2-n-1. Since x < n^2, the denominator removes exactly the prime factors of x that are at most n-1, which forces a(n) to be 1 or a single prime. So the central claim is true and the gap is fillable from the author's own materials. The paper just doesn't do it.\n\nWhat is genuinely new: the particular recurrence and the double-occurrence symmetry conjecture appear to be new, and the sequence is in the OEIS. The idea of filtering a quadratic through a gcd with a recursively-defined sequence is a real variation on Rowland and Cloitre. The observation that the prime terms seem to appear exactly twice, with the mirror formula a(p-n+1)=p, is interesting and not something I've seen before.\n\nThe soft spots are real but not fatal to the core math. Theorem 1's continued fraction proof is sketchy and the system of a_i is underdefined; the main results section reads more like a collection of OEIS tables than a worked-out theory. The \"more efficient than Rowland\" claim in the abstract is never quantified, and in any practical sense it is likely false: computing b(n) involves factorially growing values, while Rowland's recurrence is cheap. The conjecture that all primes ending in 1 or 9 appear is bold and unsupported by 10,000 terms. And the abstract states the 1-or-prime result as a finding when Section 2 calls it a conjecture.\n\nWho this is for: someone working on prime-generating recurrences or recreational number theory will find the symmetry conjecture worth chasing. As written, the paper is a computational observation with an unproved but true structural claim. It deserves a serious referee, because the missing proof is short and the author should be pushed to include it and to define efficiency properly.\n\nSend it to review, but expect major revision. The referee should ask for a proof of the gcd identity, a precise statement of the continued fraction results, and removal or quantitative support for the efficiency claim.","headline":"The real mathematical claim is true, but the paper hides the proof and overstates what is shown; worth a serious referee if only to make the author write the missing argument.","tokens_in":5068,"tokens_out":1535,"would_cite":false,"duration_ms":20283,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A41","11B83","11A55"],"pacs":[],"model":"deepseek-v4-flash","headline":"A new gcd-filtered quadratic generates only 1's and primes, more densely than Rowland's sequence.","keywords":["prime-generating sequence","gcd recurrence","Rowland sequence","continued fraction","quadratic polynomial","left factorial","integer sequence","number theory"],"falsifier":"Compute a(n) directly from the recurrence b(n) for all n up to, say, 100,000 and look for any composite a(n) > 1. Alternatively, compare gcd(n^2−n−1, b(n−3)+n b(n−4)) with gcd(n^2−n−1, (n−1)!) for the same range; any mismatch would falsify the unproved simplification and, with it, the structural explanation of the sequence.","tokens_in":4165,"feed_emoji":"🔢","tokens_out":4048,"duration_ms":43622,"temperature":0.7,"pith_summary":"This paper introduces a new integer sequence defined by dividing a quadratic polynomial by its greatest common divisor with an auxiliary recursively defined term. The author claims that every term is either 1 or a prime, that the sequence produces primes more efficiently than Rowland's well-known prime-generating recurrence, and that all primes ending in 1 or 9 appear. A continued-fraction identity connects the auxiliary recurrence to a simpler factorial form, and symmetry properties are conjectured for repeated primes. The supporting evidence is computational up to 10,000 terms, and a key simplification is asserted without proof.","feed_headline":"New sequence yields only 1s and primes, denser than Rowland's","feed_subtitle":"A quadratic divided by its gcd with a recursive denominator creates a prime-rich sequence.","key_machinery":"The core mechanism is the gcd filter: a quadratic numerator x = n^2 − n − 1 is divided by gcd(x, y), where y = b(n−3) + n b(n−4). The auxiliary sequence b is defined by a two-term linear recurrence with polynomial coefficients. The unproved identity y ≡ (n−1)! (in the sense of equal gcds) is the algebraic link that turns the denominator into a factorial-like filter. Theorem 1 establishes a finite continued-fraction identity involving b and m, which is used to generalize the construction to a family of quadratic and linear forms.","core_discovery":"The central claim is that for n ≥ 3, a(n) = (n^2 − n − 1)/gcd(n^2 − n − 1, b(n−3) + n b(n−4)) takes only the value 1 or prime values, where b(n) obeys b(n) = (n+2)(b(n−1) − b(n−2)) with b(−1)=0, b(0)=1. The paper further states that b(n−3) + n b(n−4) can be replaced by (n−1)! in the gcd without changing the result, yielding a finite continued fraction representation and a connection to the left factorial function. The sequence is claimed to be a denser prime source than Rowland's recurrence, and a symmetry conjecture predicts that each prime (except 5) appears exactly twice at positions related to p.","pith_inferences":["The unproved factorial identity may be a special case of a broader congruence between linear recurrences with polynomial coefficients and factorial functions; establishing it could unify this sequence with other gcd-based prime generators.","The claimed efficiency (only 14.2% non-primes) rests on a finite computational range; a natural extension is to investigate the asymptotic density of 1's and compare it rigorously with Rowland's sequence.","The symmetry conjecture suggests an involution on indices n ↔ p−n+1; understanding this structure might connect the sequence to prime-counting functions or modular properties of the quadratic form.","The fragility of the prime-only property under parameter changes (as seen with a_4(n)=4) raises the question of which k values preserve the property and whether the factorial identity is the precise condition."],"forward_implications":["If correct, the sequence provides a new explicit prime-generating formula: among the first 10,000 terms, 8,580 are primes and only 1,420 are the value 1, a denser prime output than Rowland's recurrence.","All primes ending in 1 or 9 would appear in the sequence, covering two complete residue classes of primes.","The symmetry property a(p−n+1)=p would allow the positions of repeated primes to be predicted directly from p, potentially simplifying the search for large primes.","The continued-fraction representation gives an alternative way to compute the denominator without iterating the b recurrence, and connects the sequence to the left factorial function.","The generalized family a_k(n) extends the construction to several quadratic and linear forms, each mostly producing 1's and primes, though one observed exception (a_4(n)=4) shows the property is not universal."],"fun_headline_variants":["Prime-only sequence denser than Rowland's","Quadratic gcd formula yields only 1s and primes","New sequence with finite continued fraction gives primes","Recursive prime generator outpaces Rowland's","Sequence of 1s and primes via gcd and recursion"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The paper's structural conclusions rely on the unproved statement that b(n−3) + n b(n−4) shares the same gcd with n^2−n−1 as (n−1)! does; if this identity fails for some n, the sequence's prime-only behavior and the continued-fraction claims are not established.","fun_headline_variants_meta":{"raw":{"variants":["Prime-only sequence denser than Rowland's","Quadratic gcd formula yields only 1s and primes","New sequence with finite continued fraction gives primes","Recursive prime generator outpaces Rowland's","Sequence of 1s and primes via gcd and recursion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000211,"raw_usage":{"total_tokens":1215,"prompt_tokens":672,"completion_tokens":543,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":416,"completion_tokens_details":{"reasoning_tokens":469}},"tokens_in":416,"tokens_out":543,"duration_ms":5654,"temperature":1.0,"reasoning_tokens":469,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T19:26:14.210852+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute a(n) directly from the recurrence b(n) for all n up to, say, 100,000 and look for any composite a(n) > 1. Alternatively, compare gcd(n^2−n−1, b(n−3)+n b(n−4)) with gcd(n^2−n−1, (n−1)!) for the same range; any mismatch would falsify the unproved simplification and, with it, the structural explanation of the sequence.","supporting_citations":[],"review_version":1}