{"id":"af00a82b-6b3a-4d73-944e-5cb804ce359f","arxiv_id":"2502.03513","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Goldbach's 'other other' conjecture, that every a with a^2+1 prime is a sum of two such a's, is verified computationally up to a ≤ 2.5×10^14.","lead":"The authors computed every prime of the form m^2+1 up to 6.25×10^28 and checked Goldbach's obscure 1742 conjecture that each such prime is a sum of two primes of the same form. The work extends prior tables by a factor of about 6000 and introduces a new search statistic, Goldbach champions, plus conditional results about how far back one must look for such sums.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The real load-bearing gap is the unarchived 30 TB sieve output; the reader's Lemma 1 mod-2 objection is invalid because the product is nonzero at x=0 over F2.","rationale":"The Reader's verdict is CONDITIONAL, citing both Lemma 1 and unarchived data. I checked Lemma 1 carefully and found the mod-2 objection incorrect: a product of polynomials can vanish at x=1 without being identically zero over F2, and at x=0 the value is 1. The proof's choice b even actually works. Proposition 2's apparently puzzling pigeonhole step also survives scrutiny because the b_i are excluded from A, so small-d representations are exactly the 'other d' values. What remains is the auditability of the computational centerpiece. The paper tables 5.34 x 10^12 primes and claims a verification beyond all previous bounds, but the output is not archived and only sample code is posted. The counts match the Bateman-Horn heuristic extremely well, which strongly suggests the sieve is correct, but it does not prove completeness at every segment boundary. This is a sound basis for a conditional verdict rather than acceptance; it does not justify rejection. The Reader's Lemma 1 attack should not be relied on, but the conditional verdict itself is unchanged.","tokens_in":7663,"tokens_out":26084,"duration_ms":235880,"concrete_test":"Regenerate a fixed random segment of the claimed m^2+1 prime list with an independent implementation (e.g., all m with 2.49 x 10^14 - 10^6 <= m <= 2.49 x 10^14, using a different sieve and root-of-minus-one routine) and compare the exact set and count against a checksummed extract provided by the authors; if the authors cannot provide such an extract, or if the sets differ, the verification claim is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The conditional results withstand the reader's stated objection. Lemma 1 chooses all b_i congruent to a single b modulo every prime p <= 2k, with b even. Modulo 2, each f_i(x) = (x-b_i)^2+1 reduces to x^2+1, which equals 1 at x=0 and 0 at x=1; the product is therefore not identically zero over F2. For p > 2k, the product has degree 2k < p and is not the zero polynomial, so the Bunyakovsky condition holds. The pigeonhole step in Proposition 2 is also defensible once one notes that the b_i are not in A: if no 'other' d < b_{k-1} occurs, the top k-1 elements of A below m lie inside (m-b_{k-1}, m), so any representation with j(m) < k would force an 'other' d. The genuinely load-bearing weakness is empirical: the verification of Conjecture 1 up to 6.25 x 10^28 rests on an unarchived 30-terabyte list and sample code only. No reader can check segment boundaries, root computations, or completeness, and a localized sieve bug could invalidate the headline while leaving the aggregate counts in Table 1 essentially unchanged. This is a reproducibility gap, not an internal inconsistency, and it is what should keep the verdict conditional.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper describes a new computation of all primes of the form m^2+1 below 6.25×10^28, extending previous results of Wolf and Gerbicz from 10^25. Using the resulting list, the authors claim to verify Goldbach's 'other other' conjecture (every a>1 with a^2+1 prime is a sum of two such elements) up to a ≤ 2.5×10^14. They introduce a statistic j(a_n), the 'Goldbach champion' index, and prove conditional results about its behavior: under Schinzel's Hypothesis H, j(a_n)>1 infinitely often and lim inf j(a_n)=1; under the Bateman-Horn conjecture, lim sup j(a_n)=∞.","tokens_in":7881,"tokens_out":39039,"duration_ms":311786,"significance":"If the computational verification is correct, it represents a substantial extension of known data and provides strong empirical support for a conjecture of Goldbach from 1742. The champion statistic j(a_n) is a natural and potentially reusable tool for such verifications. The conditional theorems give interesting extreme behavior, though they depend on standard but unproven conjectures. The paper's main weakness is the lack of a complete algorithmic description for the verification and the sieve's handling of evenness, which currently prevent independent verification of the headline computational claim.","major_comments":[{"comment":"The third-sieve primality criterion 'x^2+1 is prime if and only if x is not a square root of −1 modulo any of the primes in the second list' is false as stated for odd x > 1. For such x, x^2+1 is even and composite, but the second list contains only primes ≡ 1 (mod 4); if all odd prime divisors exceed B^{1/2} (for example x=9, B=100 gives 82 = 2·41 with 41 > 10), x is not sieved out. The description should restrict to even x (or include the prime 2) and explain how the case x=1 is handled. This is load-bearing for the completeness of the computation.","section":"Section 2"},{"comment":"The paper claims to have verified Conjecture 1 for all a ≤ 2.5×10^14, i.e., for all n ≤ π_q(6.25×10^28) ≈ 5.34×10^12, but it gives no description of the algorithm that made this feasible. In particular, it does not state how membership in A was tested, how j(a_n) was computed for every n, or what the largest value of j(a_n) in the full range was. The appendix lists champions only up to n ≈ 6.3×10^10, so it does not by itself show that the search terminates for the remaining n. The authors should provide a detailed algorithmic description and, ideally, the full verification code.","section":"Section 4"},{"comment":"In the proof by contradiction, the pigeonhole step asserts that for some fixed d, the number of m ≤ y with all k+1 polynomials prime is 'asymptotically c' y/log^2 y'. This is not the correct order for general k; the lower bound obtained from a set of size ~ y/log^k y is c' y/log^k y along a subsequence, not y/log^2 y. The contradiction with Bateman-Horn's ~ y/log^{k+1} y should be stated with this corrected bound. As printed, the claimed estimate is wrong and for k=1 would not contradict the Bateman-Horn asymptotic.","section":"Proposition 2"},{"comment":"The proof that 65y+1 and 65y+9 are consecutive elements of A only checks the intermediate values 65y+3, 65y+5, and 65y+7. It does not address 65y+2, 65y+4, 65y+6, and 65y+8. When f_2(y) is prime, 65y+9 is even, so y is odd and these four unlisted values are odd, making their squares plus one even and greater than 2; the proof should state this. As written, the consecutiveness claim is not fully justified.","section":"Proposition 1"}],"minor_comments":[{"comment":"The statement writes f_i = x^2 − b_i, but the context and proof require f_i(x) = (x − b_i)^2 + 1; the typo should be corrected.","section":"Lemma 1"},{"comment":"The proof invokes Lemma 1 to assert that x^2+1 and (x−2)^2+1 satisfy the Bunyakovsky condition, but Lemma 1 only guarantees existence of some b_i and does not directly produce this specific pair; the condition should be verified directly for these two polynomials.","section":"Proposition 3"},{"comment":"The third column appears to use log base 2, while the asymptotic formula in Section 4 uses the natural logarithm; the base of the logarithm in the table should be specified.","section":"Appendix A"},{"comment":"The set of 'other' d values in the pigeonhole argument is not explicitly defined; it should be stated as the finite set {d ∈ (0, b_{k-1}) : d ≠ b_i}.","section":"Proposition 2"},{"comment":"The paper states that the appendix contains all champions for a_n < 2.5×10^14, but the table stops at n ≈ 6.3×10^10 while the verification range extends to n ≈ 5.3×10^12; if the table is complete, this should be said explicitly, and the maximum j(a_n) over the full range should be reported.","section":"Section 4"},{"comment":"The inequality 'B > x^2+1 > B^{1/2}' should be clarified: the lower bound B^{1/2} is used to ensure all prime factors of composite x^2+1 lie in the second sieve's range, but the treatment of values x^2+1 ≤ B^{1/2} is not described.","section":"Section 2"}],"recommendation":"major_revision","confidential_remarks":"The headline contribution is computational, and the two main issues are the incomplete description of the verification algorithm and the third-sieve criterion's failure for odd x. These are fixable with additional detail. The conditional theorems are interesting but secondary; after the computational claims are made reproducible and the Proposition 1/2 proof gaps are closed, the paper would be suitable for publication. The 30 TB dataset cannot be archived, but the authors should release the full code and describe internal consistency checks."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper is better than the reader's report makes it look. The mod-2 objection to Lemma 1 does not hold up: with all b_i chosen even, each f_i(0) ≡ 1 mod 2, so the product is not identically zero over F_2, and for p > 2k the degree is less than p, so the Bunyakovsky condition is satisfied. The CRT construction is sound. Proposition 2's pigeonhole step is terse but the contradiction is real: if all but finitely many Bateman-Horn m's had some extra d, then the union of finitely many k+1-tuple sets, each of size O(y/log^{k+1} y), would cover a set of size Θ(y/log^k y), impossible. Proposition 3 doesn't even need the lemma; the two polynomials can be checked directly. So the conditional results stand.\n\nWhat is genuinely new: the extension of the m^2+1 prime list from 10^25 to 6.25×10^28, the verification of Goldbach's 'other other' conjecture up to a ≤ 2.5×10^14, and the introduction of Goldbach champions j(a_n) with the champion table. The triple-sieve exposition is clear, and the complexity analysis is careful. The counts in Table 1 track the Hardy-Littlewood prediction unusually well, which is a good sign for the computation.\n\nThe real soft spot is reproducibility. The full sieve output is 30 TB and unarchived, and only sample code is released. A referee cannot check that every a ≤ 2.5×10^14 satisfies the Goldbach property. A localized bug in the sieve could break the verification while barely moving the aggregate counts. The verification search itself is also under-described. I don't suspect dishonesty, but it's a genuine gap.\n\nMinor issues: the statement of Lemma 1 has a typo ('x2 − bi'), and the proof of Proposition 2 is written in a way that invites confusion around the constants. These are fixable.\n\nThe paper is for computational number theorists who care about primes of the form m^2+1 and, more generally, prime values of quadratic polynomials. It's not a breakthrough, but it's honest progress in a niche area. Send it to a serious referee. The referee should push on the data-availability question and on making the verification algorithm explicit, but the conditional theorems should survive scrutiny.","headline":"The conditional theorems are actually fine once you look past the mod-2 red herring; the real reason to keep this conditional is the unarchived 30 TB.","tokens_in":8462,"tokens_out":9840,"would_cite":false,"duration_ms":78291,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N32","11P32"],"pacs":[],"model":"deepseek-v4-flash","headline":"Goldbach's 1742 conjecture verified to 2.5×10^14","keywords":["Goldbach's other other conjecture","primes of the form m^2+1","sums of two squares","Bateman-Horn conjecture","Schinzel's Hypothesis H","sieve of Eratosthenes","Goldbach champions","prime counting"],"falsifier":"Compute the product of the polynomials $(x-b_i)^2+1$ modulo 2 using the paper's 'all $b_i$ in one residue class' construction; if that product is identically zero over $\\mathbb{F}_2$, then the lemma's proof does not establish the Bunyakovsky condition, and Propositions 2 and 3 lack support. Separately, an independent segmented-sieve computation of $\\pi_q(6.25\\times10^{28})$ that disagrees with the reported $5{,}342{,}656{,}862{,}803$ would falsify the enumeration.","tokens_in":7413,"feed_emoji":"🔢","tokens_out":14269,"duration_ms":113640,"temperature":0.7,"pith_summary":"The paper extends the computation of primes of the form $m^2+1$ from $10^{25}$ to $6.25\\times10^{28}$, producing a list of $5{,}342{,}656{,}862{,}803$ such primes. Using that list, it verifies Goldbach's 'other other conjecture' up to $a=2.5\\times10^{14}$: every $a>1$ with $a^2+1$ prime in that range is a sum $b+c$ with $b,c$ also of the form $x^2+1$ prime. To organize the verification, the authors define a statistic $j(a_n)$, the number of previous elements one must look back before finding a Goldbach representation, and tabulate record values ('Goldbach champions'). They also prove conditional statements about $j$: under Schinzel's Hypothesis H it exceeds 1 infinitely often and equals 1 infinitely often, and under the Bateman-Horn conjecture it is unbounded. If these results hold, the computation supplies the strongest numerical evidence to date for a conjecture Goldbach sent to Euler in 1742.","feed_headline":"Goldbach's 1742 conjecture verified to 2.5×10^14","feed_subtitle":"A 30-terabyte sieve lists every prime of the form m^2+1 below 6.25×10^28 and checks each Goldbach representation.","key_machinery":"The load-bearing computational object is a three-stage sieve. First, the sieve of Eratosthenes generates primes up to $B^{1/4}$. Second, a segmented sieve produces all primes $p\\equiv1\\pmod4$ up to $B^{1/2}$ and stores, for each, the two square roots of $-1$ modulo $p$. Third, a segmented sieve over $x\\le B^{1/2}$ uses those roots to mark every $x$ for which $x^2+1$ is divisible by such a prime, so the unmarked $x$ give exactly the primes of the form $m^2+1$. The theoretical machinery is the statistic $j(a_n)$, the smallest $i$ such that $a_n-a_{n-i}\\in A$, with 'champion' records; the conditional proofs construct tuples of shifted polynomials $(x-b_i)^2+1$ and invoke Hypothesis H or the Bateman-Horn conjecture to force simultaneous prime values.","core_discovery":"The paper's central computational discovery is that the set $A=\\{a: a^2+1\\text{ is prime}\\}$ has the Goldbach property through $a=2.5\\times10^{14}$: every element of $A$ with $11$ infinitely often and $j(a_n)=1$ infinitely often; under Bateman-Horn, $\\limsup j(a_n)=\\infty$.","pith_inferences":["The same three-stage sieve could likely be adapted to other quadratic polynomials $x^2+D$ or to cyclotomic polynomials, as the authors say they plan; the practical bottleneck is memory bandwidth, not the asymptotic sieve cost.","The champion statistic $j(a_n)$ behaves like a merit for Goldbach representations, analogous to prime-gap merit, and the slow growth of its record values suggests typical look-back is far smaller than the champion values.","If the conditional lemma is repaired, a Bateman-Horn-style counting argument might give a quantitative density for the set of $a$ with $j(a_n)=k$, rather than only unboundedness.","Because the 30-terabyte list is not archived, independent verification of the reported count, or of checksums of the list, would substantially strengthen the empirical claim; the published tables of counts and champions are the only directly checkable artifacts."],"forward_implications":["Goldbach's other other conjecture is now verified for all $a\\le 2.5\\times10^{14}$, extending the known range of a conjecture that dates to 1742.","The count $\\pi_q(6.25\\times10^{28})$ continues the sequence of known counts and matches the conjectured asymptotic $\\frac{C_q}{2}\\operatorname{li}(\\sqrt{x})$ to within a few parts in $10^7$ at the new bound.","Under the Bateman-Horn conjecture, the look-back statistic $j(a_n)$ is unbounded: for every fixed $k$, infinitely many elements $a_n$ of $A$ have no Goldbach partner among the previous $k$ elements.","Under Schinzel's Hypothesis H, $j(a_n)=1$ infinitely often, so the conjecture is true infinitely often, while $j(a_n)>1$ infinitely often, so the nearest previous element is not always a partner.","The champion table provides concrete record values of how far back one must look, with $j(a_n)/\\log n$ reaching about $5$ at the largest recorded champion."],"supporting_citations":[{"why":"Supplies the prior computation of primes $m^2+1<10^{20}$, the notation $\\pi_q(x)$, and the table structure that this paper extends.","marker":"[Wol13]"},{"why":"Extends the count of such primes to $10^{25}$, the baseline the new computation continues.","marker":"[WG10]"},{"why":"Hardy and Littlewood's Conjecture E provides the asymptotic $f(x)$ and the constant $C_q$ used to compare the computed counts.","marker":"[HL23]"},{"why":"The Bateman-Horn Conjecture is the assumption behind the claim that $\\limsup j(a_n)=\\infty$.","marker":"[BH62]"},{"why":"Schinzel's Hypothesis H is the assumption behind the propositions that $j(a_n)>1$ infinitely often and $j(a_n)=1$ infinitely often.","marker":"[SS58]"},{"why":"Provides the sieve complexity estimate $O(A\\log\\log P+P^{1/2}/\\log P)$ used to analyze the triple sieve's running time.","marker":"[CP05]"},{"why":"Contains Goldbach's 1742 letter stating the conjecture that the paper verifies computationally.","marker":"[Fus68]"},{"why":"Documents the counterexamples to the related 'other' conjecture, clarifying why the present conjecture is a distinct claim.","marker":"[Hod93]"}],"fun_headline_variants":["Goldbach's 1742 conjecture verified at 2.5e14","Goldbach champions emerge in 30-TB prime census","5.3 trillion m^2+1 primes confirm Goldbach's other claim","30-TB sieve proves Goldbach's 1742 side claim","Goldbach's other conjecture held to 2.5e14 via 30-TB sieve"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The computational claim rests on the completeness and correctness of the unarchived 30-terabyte sieve output, and Propositions 2 and 3 rest on the lemma that one can always choose $k$ shifts $b_i$ so that the polynomials $(x-b_i)^2+1$ satisfy the required no-common-prime-divisor condition; the proof's construction for that lemma does not work modulo 2.","fun_headline_variants_meta":{"raw":{"variants":["Goldbach's 1742 conjecture verified at 2.5e14","Goldbach champions emerge in 30-TB prime census","5.3 trillion m^2+1 primes confirm Goldbach's other claim","30-TB sieve proves Goldbach's 1742 side claim","Goldbach's other conjecture held to 2.5e14 via 30-TB sieve"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000984,"raw_usage":{"total_tokens":4144,"prompt_tokens":883,"completion_tokens":3261,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":499,"completion_tokens_details":{"reasoning_tokens":3175}},"tokens_in":499,"tokens_out":3261,"duration_ms":22299,"temperature":1.0,"reasoning_tokens":3175,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T04:43:33.607057+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the product of the polynomials $(x-b_i)^2+1$ modulo 2 using the paper's 'all $b_i$ in one residue class' construction; if that product is identically zero over $\\mathbb{F}_2$, then the lemma's proof does not establish the Bunyakovsky condition, and Propositions 2 and 3 lack support. Separately, an independent segmented-sieve computation of $\\pi_q(6.25\\times10^{28})$ that disagrees with the reported $5{,}342{,}656{,}862{,}803$ would falsify the enumeration.","supporting_citations":[],"review_version":1}