{"id":"eb2073c1-afd5-4e97-a336-a0b0cc8c1c3a","arxiv_id":"2508.02821","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":3.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":2,"one_line_summary":"A parametrized quadratic polynomial family using Heegner numbers is claimed to optimize prime generation, conditionally on the Bateman-Horn conjecture.","lead":"This paper proposes a family of quadratic polynomials that can be tuned to produce many prime numbers, built by shifting the classic Euler-Rabinowitsch polynomials and using special numbers called Heegner numbers. The authors argue, using the unproved Bateman-Horn conjecture, that the family can achieve high prime density.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"For H=1 and H=2, the coefficient ((2Zk−1)^2+H)/4 is non-integral, so the proposed polynomial has rational coefficients and cannot generate primes.","rationale":"The abstract's core claim is the construction of a parametrized prime-generating family. A necessary condition is that the polynomial takes integer values at integer inputs. Because (2Zk−1)^2 is always 1 mod 4, the constant term is integral iff H ≡ 3 mod 4. Two of the nine Heegner numbers, 1 and 2, violate this condition, and explicit substitution shows the family as written has rational coefficients for those H. This concern is load-bearing because it precedes any prime-density estimate: Bateman-Horn and prime-counting functions cannot be meaningfully applied to a rational-valued function in this setting. The reader's emphasis on the unproved Bateman-Horn conjecture is a valid but secondary issue. The construction can likely be repaired by restricting H to the Heegner numbers congruent to 3 modulo 4, so a conditional verdict is more actionable than outright rejection; however, full verification of the density claim still requires the missing text. Agreement with the reader is partial because both question the family's applicability, but the identified mechanism here—non-integrality—is different from reliance on an unproved conjecture.","tokens_in":718,"tokens_out":7105,"duration_ms":80166,"concrete_test":"Substitute Zk=1 (so B=1) with H=1 and H=2 into f_{Z,k,H}(n). The constant terms are (1+1)/4=1/2 and (1+2)/4=3/4, so f(0) is non-integral and the polynomial is not integer-valued over the integers. This single computation settles whether the definition as written is consistent for the full Heegner list; if it fails, check whether restricting to H ∈ {3,7,11,19,43,67,163} removes the obstruction and whether the claimed high prime density still holds on that restricted set.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"For the family to be a prime-generating polynomial, f_{Z,k,H}(n) must be integer-valued on integer n. Write B=2Zk−1; for every integer Zk, B is odd, so B^2 ≡ 1 (mod 4). The constant term is c=(B^2+H)/4, and c is an integer if and only if H ≡ 3 (mod 4). The set of Heegner numbers is {1,2,3,7,11,19,43,67,163}. Taking Zk=1 (B=1), H=1 gives c=(1+1)/4=1/2, and H=2 gives c=(1+2)/4=3/4. Thus f(n)=n^2−n+1/2 and f(n)=n^2−n+3/4, which are not integer-valued at integer n. Therefore the abstract's assertion that H belongs to the set of Heegner numbers cannot hold for all Heegner numbers if f is meant to be a polynomial generating primes. This is an internal consistency problem in the central object, independent of the Bateman-Horn heuristic. If the intended domain is only the Heegner numbers congruent to 3 modulo 4, or if the authors use a different convention making the coefficients integral, the statement must be revised and the density estimates recomputed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript (or the abstract under review) introduces a family of quadratic polynomials f_{Z,k,H}(n) = n^2 - (2Zk - 1)n + ((2Zk-1)^2 + H)/4, where Zk is a nonnegative integer and H is claimed to be a Heegner number. The author asserts a close relationship to Euler-Rabinowitsch polynomials and claims, on the basis of the Bateman-Horn conjecture and prime-counting functions, that the family can be optimized to generate a high density of primes. The abstract offers no derivation, no explicit optimization procedure, and no numerical evidence; the only technical content is the family definition and the heuristic appeal to Bateman-Horn.","tokens_in":973,"tokens_out":5085,"duration_ms":56646,"significance":"If the central construction is corrected and the claimed density results are established, the family could provide a unifying framework for known prime-generating quadratics and a systematic way to search for record-holding polynomials. The connection to Heegner numbers is a plausible source of algebraic structure. However, the abstract as written is internally inconsistent because the polynomial is not integer-valued for two of the nine Heegner numbers, and no evidence is supplied for the optimization claim. The significance of the contribution therefore cannot be assessed from the current manuscript.","major_comments":[{"comment":"The family as defined in the abstract is not integer-valued for all Heegner numbers H. For every integer Zk, B = 2Zk - 1 is odd, so B^2 ≡ 1 mod 4. Consequently the constant term (B^2 + H)/4 is an integer only when H ≡ 3 mod 4. Since the Heegner numbers include 1 and 2, the choices H = 1 and H = 2 give f(n) = n^2 - n + 1/2 and f(n) = n^2 - n + 3/4, respectively, which are not integers for any integer n and therefore cannot generate primes. The statement that H belongs to the set of Heegner numbers must be restricted to H ≡ 3 mod 4, or the definition must be modified, and all subsequent density computations must be redone for the admissible set.","section":"Abstract, definition of f_{Z,k,H}"},{"comment":"The abstract claims that the Bateman-Horn conjecture and prime-counting functions are used to 'demonstrate' that the family can be optimized to generate a high density of primes. No details of this demonstration appear in the text provided. The reader cannot verify that the Bateman-Horn constant has been computed for this family, that the optimization over Zk and H has actually been performed, or what the resulting density is. If the full paper contains these computations, they must be made explicit; if not, the word 'demonstrate' is too strong and should be replaced by 'suggest' or 'provide heuristic evidence for.'","section":"Abstract, Bateman-Horn claim"},{"comment":"The abstract states that the form is 'closely related to the Euler-Rabinowitsch polynomials through specific substitutions.' Since Euler-Rabinowitsch polynomials of the form n^2 + n + A are already known to be prime-rich when A is a Heegner number (or related to the class number), the present family may simply be a reparameterization of the same objects. The abstract does not specify the substitution or explain how the two-parameter family provides new algebraic tuning that is not already captured by the classical Euler-Rabinowitsch form. Please state the precise relation and identify which aspect of the construction is genuinely new.","section":"Abstract, novelty and relation to Euler-Rabinowitsch polynomials"}],"minor_comments":[{"comment":"The notation 'Zk' is unusual; presumably it denotes a single symbol such as Z_k or z_k. Please clarify whether it is a product Z·k or a single parameter, and use standard subscript notation.","section":"Abstract, notation"},{"comment":"The term 'Heegner number' should be defined or referenced, as not all readers of analytic number theory will be familiar with the list {1, 2, 3, 7, 11, 19, 43, 67, 163} and its connection to class number one.","section":"Abstract, terminology"},{"comment":"The phrase 'high density of primes' is not quantified. The abstract should specify the expected number of primes for n ≤ N or, at a minimum, state the leading constant from the Bateman-Horn conjecture so that the claimed optimization can be evaluated.","section":"Abstract, quantitative claim"},{"comment":"The statement that the work has 'potential applications in cryptography and signal processing' is unsupported by any argument or known mechanism. Unless the full paper supplies a concrete connection, this sentence should be removed or substantially softened.","section":"Abstract, applications"}],"recommendation":"major_revision","confidential_remarks":"The submitted text is an abstract only, so a full technical evaluation is not possible. The most serious issue is a definitive error in the stated definition: for H = 1 and H = 2 the polynomial is not integer-valued, contradicting the claim that H ranges over all Heegner numbers. This is fixable by restricting to H ≡ 3 mod 4, but it means the central object as stated is incorrect and all density estimates derived from it need to be revisited. The lack of any derivation or numerical evidence in the abstract is also a concern, though a full paper may remedy it. I recommend major revision with the understanding that the full manuscript must be re-reviewed once the definition is corrected and the claimed computations are actually supplied."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the abstract's family is not integer-valued for H=1 and H=2. With Zk=1, B=2Zk-1=1, and the constant term is (1+H)/4, which is 1/2 for H=1 and 3/4 for H=2. Since H is claimed to range over all Heegner numbers, the definition as written cannot produce primes for those two values. That's a load-bearing problem, not a typo. Unless the full text restricts H to Heegner numbers congruent to 3 mod 4, the paper needs a major fix.\n\nWhat's genuinely here: the Zk parametrization is a small extension of the Euler-Rabinowitsch idea, and the authors are honest that it's 'closely related through substitutions.' The link to Heegner numbers is real but already known; the prime-richness is inherited from them. Using Bateman-Horn to estimate prime density is a standard tool, not a proof.\n\nThe novelty is thin. If you write B=2Zk-1, the polynomial becomes n^2 - Bn + (B^2+H)/4, which is just a shifted and scaled version of x^2+H. The Heegner numbers are finite, so the family doesn't open a new direction beyond what Euler-Rabinowitsch already gives. The claimed optimization over Zk is just choosing a shift; it doesn't change the discriminant.\n\nI can't check the density computations because only the abstract is available. But the integrality issue is visible from the abstract alone, and it's the first thing a referee would hit. If the full text clears that up and shows explicit examples, the paper could be a minor contribution. As it stands, the abstract doesn't support the central claim.\n\nI would not send this to peer review. It's not ready; the definition is defective as stated. The authors should fix the H restriction and recompute any density estimates. Then a short note might be refereeable, but I wouldn't put effort into it until the integrality question is resolved.","headline":"The abstract's polynomial is not integer-valued for two of the Heegner numbers, so as stated the family cannot generate primes—that's a disqualifying flaw unless the full text restricts H.","tokens_in":1516,"tokens_out":2920,"would_cite":false,"duration_ms":30087,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N32","11N05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that a two-parameter quadratic family built from Heegner numbers can be tuned to produce a high density of primes, with the density estimate resting on the Bateman-Horn conjecture.","keywords":["prime-generating polynomials","Heegner numbers","Bateman-Horn conjecture","Euler-Rabinowitsch polynomials","quadratic number fields","prime distribution","analytic number theory","structured primes"],"falsifier":"Take a fixed large Heegner number such as $H=163$, compute the actual prime counts of $f_{Z,k,H}(n)$ for increasing ranges of $n$ and a sweep of $Zk$, and compare the limiting trend to the Bateman-Horn prediction; a systematic shortfall, or an optimized $(Z,k)$ whose observed density does not beat generic quadratics, would refute the central claim.","tokens_in":514,"feed_emoji":"🔢","tokens_out":8099,"duration_ms":74995,"temperature":0.7,"pith_summary":"This paper introduces a family of quadratic polynomials $f_{Z,k,H}(n)=n^2-(2Zk-1)n+\\frac{(2Zk-1)^2+H}{4}$, where $Zk$ is a nonnegative integer and $H$ runs through the Heegner numbers. The authors argue that this family generalizes the classical Euler-Rabinowitsch prime-generating quadratics, and that by choosing the parameters appropriately the polynomial yields an unusually high density of prime values. They support this claim using the Bateman-Horn conjecture, which predicts how many primes a polynomial of this kind should produce, and prime-counting functions. If correct, the result would give a tunable, structured source of primes and would indicate that Heegner numbers measurably influence prime distribution in quadratic sequences.","feed_headline":"Heegner-number quadratics claim high-density prime output","feed_subtitle":"A tunable quadratic, guided by Heegner numbers, is said to run unusually long on primes before failing.","key_machinery":"The central object is the polynomial family $f_{Z,k,H}(n)=n^2-(2Zk-1)n+\\frac{(2Zk-1)^2+H}{4}$, with $H$ a Heegner number (one of the nine integers whose imaginary quadratic field has class number one) and $Zk$ a nonnegative integer. The parameter $Zk$ moves the vertex and tunes the constant term, while $H$ enters the discriminant-like structure that ties the family to Euler-Rabinowitsch polynomials. The quantitative engine is the Bateman-Horn conjecture, which supplies a predicted asymptotic density of primes for such polynomials; prime-counting functions are then used to compare or optimize the parameter choices.","core_discovery":"The central claim is that $f_{Z,k,H}(n)=n^2-(2Zk-1)n+\\frac{(2Zk-1)^2+H}{4}$ is a prime-generating quadratic whose behavior can be controlled by the pair $(Z,k)$ while $H$ ranges over the Heegner numbers. The paper states that through specific substitutions this form is closely related to the Euler-Rabinowitsch polynomials, so it both recovers known prime-rich quadratics and adds extra tuning freedom. Using the Bateman-Horn conjecture and prime-counting functions, the paper asserts that the family can be optimized to generate a high density of primes, and takes this as evidence that Heegner numbers have a concrete impact on the distribution of primes in quadratic sequences.","pith_inferences":["Inference: the parametrization may turn out to be a reparameterization of the Euler-Rabinowitsch family, in which case the claimed high density would be a known property in new clothing rather than a genuinely new phenomenon.","Inference: even with no proof of Bateman-Horn, a finite numerical sweep over $Zk$ and $H$ could test whether certain Heegner numbers consistently yield longer prime runs than others, giving a partial check of the paper's heuristic claim.","Inference: for cryptographic use the primes would need to be large and effectively unpredictable, and a simple quadratic sequence rarely offers both; the paper's abstract does not address that gap."],"forward_implications":["The family is claimed to contain or reproduce the classical Euler-Rabinowitsch behavior, so its prime-rich runs extend a long-studied phenomenon rather than appearing in isolation.","If Bateman-Horn applies, choosing different Heegner numbers and different $(Z,k)$ pairs gives a way to tune a quadratic to produce primes with higher density than generic quadratics.","The analysis suggests that Heegner numbers, not just the discriminant, affect how long a quadratic can avoid composite values.","A reliable tunable quadratic would offer a simple source of structured primes, which the paper suggests could matter for cryptography and signal processing."],"supporting_citations":[],"fun_headline_variants":["Heegner-number quadratics tune prime-dense runs","Quadratic primes get Heegner-number tuning","Heegner-guided quadratics for prime-rich outputs","New quadratic family tunes primes via Heegner numbers","Heegner numbers refine Euler-Rabinowitsch quadratics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The density estimate depends on the Bateman-Horn conjecture, an unproved statement about how often a polynomial takes prime values, and if that conjecture fails for this family the claimed high prime density is not supported.","fun_headline_variants_meta":{"raw":{"variants":["Heegner-number quadratics tune prime-dense runs","Quadratic primes get Heegner-number tuning","Heegner-guided quadratics for prime-rich outputs","New quadratic family tunes primes via Heegner numbers","Heegner numbers refine Euler-Rabinowitsch quadratics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000577,"raw_usage":{"total_tokens":2675,"prompt_tokens":852,"completion_tokens":1823,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":468,"completion_tokens_details":{"reasoning_tokens":1742}},"tokens_in":468,"tokens_out":1823,"duration_ms":15844,"temperature":1.0,"reasoning_tokens":1742,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T04:50:21.940511+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a fixed large Heegner number such as $H=163$, compute the actual prime counts of $f_{Z,k,H}(n)$ for increasing ranges of $n$ and a sweep of $Zk$, and compare the limiting trend to the Bateman-Horn prediction; a systematic shortfall, or an optimized $(Z,k)$ whose observed density does not beat generic quadratics, would refute the central claim.","supporting_citations":[],"review_version":1}