Primes of the Form $m^2+1$ and Goldbach's `Other Other' Conjecture

Hester Graves; Jon Grantham

REVIEW 4 major objections 6 minor 11 references

Primes of the Form $m^2+1$ and Goldbach's `Other Other' Conjecture

T0 review · 4 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Goldbach's 1742 conjecture verified to 2.5×10^14

desk verdict 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. read the letter →

arxiv 2502.03513 v1 pith:HKRDFTMA submitted 2025-02-05 math.NT

Jon Grantham , Hester Graves This is my paper · ORCID

classification math.NT MSC 11N3211P32

keywords Goldbach'sotherconjectureprimesoftheformm^2+1sumstwosquaresBateman-HornSchinzel'sHypothesisHsieveEratosthenesGoldbachchampionsprimecounting

topics Goldbach's Conjecture

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $1<a\le 2.5\times10^{14}$ is the sum of two elements of $A$. This is established by a segmented three-stage sieve that produces the complete list of primes $m^2+1<6.25\times10^{28}$, a list occupying more than 30 terabytes. The paper reports the count $\pi_q(6.25\times10^{28})=5{,}342{,}656{,}862{,}803$, extending earlier tables, and observes that the counts track the Hardy-Littlewood/Bateman-Horn heuristic $g(x)=\frac{C_q}{2}\operatorname{li}(\sqrt{x})$ closely at the new bound. The theoretical half introduces 'Goldbach champions' $a_n$ for which the look-back statistic $j(a_n)$ sets a new record, and proves conditional results: under Hypothesis H, $j(a_n)>1$ infinitely often and $j(a_n)=1$ infinitely often; under Bateman-Horn, $\limsup j(a_n)=\infty$.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What 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.

The 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.

Minor 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.

The 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.

Referee Report

4 major / 6 minor

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)=∞.

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 (4)

[Section 2] 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 4] 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.
[Proposition 2] 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.
[Proposition 1] 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.

minor comments (6)

[Lemma 1] 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.
[Proposition 3] 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.
[Appendix A] 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.
[Proposition 2] 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 4] 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 2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the computational verification is a direct exhaustive check and the conditional results assume external conjectures; no fitted parameter or self-referential input is renamed as a prediction.

full rationale

The paper's central claim is a computational verification: it enumerates all primes of the form m^2+1 below 6.25x10^28 and checks Goldbach's other other conjecture by testing, for each a_n, whether a_n - a_n-i lies in the same computed set A. This is a direct exhaustive check, not a fitted model predicting its own calibration data. The Hardy-Littlewood constant C_q is quoted from [HL23] and used only for heuristic comparison in Table 1; it is not fitted to the newly computed counts, so the ratio columns are comparisons against an external conjectured density, not a prediction forced by construction. The conditional results in Propositions 1-3 explicitly assume Schinzel's Hypothesis H or the Bateman-Horn Conjecture, which is an assumption burden rather than circular derivation; Lemma 1 is a standard CRT and degree argument used to satisfy the Bunyakovsky condition, and its content does not presuppose Goldbach's other other conjecture. The reference list contains no papers by Grantham or Graves, so there is no load-bearing self-citation. The unarchived 30-terabyte sieve output and sample-only code repository raise reproducibility concerns, but those are not instances of circular reasoning. Accordingly, no circular step is identifiable, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; the only numerical constant Cq is taken from the Hardy-Littlewood conjecture and used for heuristics. The axioms are standard unproved prime conjectures plus the correctness of the computational pipeline. Lemma 1, which the paper tries to prove, is not included as an axiom because it is a proof step, and it is flawed as discussed in the red flags.

assumptions (4)

domain assumption Schinzel's Hypothesis H
Invoked in Proposition 1 and Proposition 3 to assert simultaneous primality of polynomial families.
domain assumption Bateman-Horn Conjecture
Invoked in Proposition 2 to count simultaneous prime values of k polynomials and to derive a contradiction from adding one more polynomial.
domain assumption Hardy-Littlewood Conjecture E for π_q(x)
Used in Section 3 and Section 4 to convert the prime count into the estimate a_n ~ (2/Cq) n log(2n/Cq); this estimate is only used for heuristics and the champions table, not for the verification itself.
domain assumption The three-sieve computation correctly and completely identifies all primes m^2+1 up to the stated bound
The verification claim depends on this; the paper provides sample code and a complexity analysis but no independent certificate or full dataset.

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Cite this review

Pith. "Pith review of Primes of the Form $m^2+1$ and Goldbach's `Other Other' Conjecture." pith.science (2026). https://pith.science/paper/HKRDFTMA

@misc{pith2026250203513,
  author       = {Pith},
  title        = {Pith review of: Primes of the Form $m^2+1$ and Goldbach's `Other Other' Conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HKRDFTMA}},
  note         = {Machine review of arXiv:2502.03513}
}

abstract

We compute all primes up to $6.25\times 10^{28}$ of the form $m^2+1$. Calculations using this list verify, up to our bound, a less famous conjecture of Goldbach. We introduce `Goldbach champions' as part of the verification process and prove conditional results about them, assuming either Schinzel's Hypothesis H or the Bateman-Horn Conjecture.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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Bateman and Roger A

Paul T. Bateman and Roger A. Horn. A heuristic asymptotic formula concerning the distribution of prime numbers. Math. Comp. , 16:363--367, 1962

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[2]

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Bateman and Roger A

Paul T. Bateman and Roger A. Horn. A heuristic asymptotic formula concerning the distribution of prime numbers. Math. Comp. , 16:363--367, 1962

work page 1962

[2] [2]

Prime Numbers: A Computational Perspective

Richard Crandall and Carl Pomerance. Prime Numbers: A Computational Perspective . Springer, New York, second edition, 2005

work page 2005

[3] [3]

Goldbach's other conjecture

Project Euler. Goldbach's other conjecture. https://projecteuler.net/problem=46. Problem 46

work page

[4] [4]

P.-H. Fuss. Correspondance math\' e matique et physique de quelques c\' e l\`ebres g\' e om\`etres du XVIII \`eme si\`ecle. T omes I , II . Pr\' e c\' e d\' e e d'une notice sur les travaux de L\' e onard Euler, tant imprim\' e s qu'in\' e dits et publi\' e e sous les auspices de l'Acad\' e mie Imp\' e riale des Sciences de Saint-P\' e tersbourg. The Sour...

work page 1968

[5] [5]

G. H. Hardy and J. E. Littlewood. Some problems of ` P artitio numerorum'; III : O n the expression of a number as a sum of primes. Acta Math. , 44(1):1--70, 1923

work page 1923

[6] [6]

A L esser- K nown G oldbach C onjecture

Laurent Hodges. A L esser- K nown G oldbach C onjecture. Math. Mag. , 66(1):45--47, 1993

work page 1993

[7] [7]

The largest prime factor of n^2+1 and improvements on subexponential ABC

Hector Pasten. The largest prime factor of n^2+1 and improvements on subexponential ABC . Invent. Math. , 236(1):373--385, 2024

work page 2024

[8] [8]

Landau's problems on primes

J\' a nos Pintz. Landau's problems on primes. J. Th\' e or. Nombres Bordeaux , 21(2):357--404, 2009

work page 2009