Goldbach's `Other Other' Conjecture (1742): must every $a>1$ with $a^2+1$ prime be a sum of two such numbers?

In a letter to Euler dated 1 October 1742, Goldbach conjectured that every $a>1$ for which $a^2+1$ is prime can be written as $b+c$ where $b^2+1$ and $c^2+1$ are also prime. Grantham and Graves (2025) resurrected this conjecture — they call it Goldbach's `Other Other' Conjecture, distinguishing it from the binary Goldbach conjecture that every even integer greater than 2 is a sum of two primes — verified it for all $a\le 2.5\times 10^{14}$ by a segmented-sieve computation (a sieve processed in bounded-memory blocks) enumerating every prime $m^2+1\le 6.25\times 10^{28}$, and proved conditional results on the backward-index function $j(a_n)$ (the smallest $i$ with $a_n-a_{n-i}$ again in $A$, where $A=\{a_1<a_2<\dots\}$ is the sequence of positive integers whose square plus one is prime) under Schinzel's Hypothesis H and the Bateman–Horn conjecture. The conjecture remains open. No implication in either direction between it and Landau's fourth problem (the infinitude of primes $m^2+1$) is known; an unconditional proof would be the first Goldbach-type additive theorem inside the prime values of a nonlinear polynomial.

Background

Let $A=\{a\ge 1: a^2+1\ \text{is prime}\}$, so $A$ begins $1,2,4,6,10,14,16,20,24,26,36,\dots$ (e.g. $4\in A$ because $17$ is prime). Whether $A$ is infinite is Landau's fourth problem, which Landau in 1912 called "unattackable at the present state of science"; it is one of the oldest open problems in number theory. The best unconditional results are far weaker than infinitude: Iwaniec (1978), using sieve methods (combinatorial procedures that bound how many values of a polynomial can avoid all small prime factors), proved that $n^2+1$ has at most two prime factors for infinitely many $n$; Pasten (2024), combining transcendence methods with modular approaches to the $ABC$ conjecture (a central unproved conjecture of Diophantine analysis relating the prime factorization of a sum to those of its summands), proved that the largest prime factor $P(n^2+1)$ satisfies $P(n^2+1)\gg (\log\log n)^2/\log\log\log n$, improving Chowla's 1934 bound of order $\log\log n$. Heuristic asymptotics for $A$ go back to Hardy and Littlewood (1923, Conjecture E): the counting function $\pi_q(x)=|\{m\le x: m^2+1\ \text{prime}\}|$ should obey $\pi_q(x)\sim C_q\sqrt{x}/\log x$ with $C_q\approx 1.3728$, and computational tables agree well (Wolf 2013; Wolf–Gerbicz to $10^{25}$; Grantham–Graves to $6.25\times 10^{28}$, recorded in OEIS A083844).

Against this background, Grantham and Graves (2025) recalled a third Goldbach conjecture, from his letter to Euler of 1 October 1742 (as documented in the Fuss edition of the Euler–Goldbach correspondence): every $a>1$ in $A$ is a sum of two elements of $A$. They call it Goldbach's Other Other' Conjecture, distinguishing it from the famous binary Goldbach conjecture (every even integer greater than 2 is a sum of two primes) and from Goldbach's other' (erroneous) conjecture that every odd composite number is a prime plus twice a square, $p+2x^2$, refuted by $5777$ and $5993$ (Hodges 1993; the Grantham–Graves preprint misprints this form as $p+x^2$). Grantham and Graves verified the conjecture for all $a\le 2.5\times 10^{14}$ by a triple segmented sieve (a segmented sieve scans the search range in consecutive blocks so that memory use stays bounded) enumerating all primes $m^2+1\le 6.25\times 10^{28}$. To study representations they enumerate $A=\{a_1<a_2<\dots\}$ and define the backward index $j(a_n)$ as the smallest $i\ge 1$ with $a_n-a_{n-i}\in A$; record values of $j$ are called Goldbach champions. Conditionally, they prove: under Schinzel's Hypothesis H (the conjecture of Schinzel and Sierpiński that finitely many integer polynomials with no fixed prime divisor — the Bunyakovsky condition — are simultaneously prime infinitely often), $\liminf j(a_n)=1$ and $j(a_n)>1$ infinitely often, so the `Other Other' conjecture holds for infinitely many $a$; under the stronger Bateman–Horn conjecture (which additionally predicts the asymptotic count of such simultaneous prime values), $\limsup j(a_n)=\infty$. Nothing unconditional is known about the growth of $j(a_n)$. Note finally that the conjecture's logical relationship to Landau's fourth problem is itself open: the closure property it demands is satisfied by finite sets (e.g. $\{1,2\}$, since $2=1+1$), so its truth would not by itself force $A$ to be infinite, and no implication in either direction between the two statements is known; the Grantham–Graves paper itself claims none.

Problem Statement

Prove or disprove Goldbach's `Other Other' Conjecture (Goldbach's letter to Euler, 1 October 1742; Conjecture 1 of Grantham–Graves, arXiv:2502.03513): let $A=\{a\ge 1: a^2+1\ \text{is prime}\}$; then every $a\in A$ with $a>1$ can be written as $a=b+c$ for some $b,c\in A$ (with $b=c$ allowed). The claim must be established (or refuted) unconditionally, not merely under Schinzel's Hypothesis H or the Bateman–Horn conjecture; the conjecture is currently verified for all $a\le 2.5\times 10^{14}$. No logical implication in either direction between this conjecture and Landau's fourth problem (the infinitude of $A$) is currently known, so neither reduces to the other by any known argument. The distinct follow-up questions raised in the same paper — an unconditional growth law or expected-value formula for the backward index $j(a_n)$, the cyclotomic generalization, and the variant representing all positive integers — are adjacent directions, not part of this problem.

References

1. Jon Grantham and Hester Graves, "Primes of the Form $m^2+1$ and Goldbach's `Other Other' Conjecture", preprint, 2025. arXiv:2502.03513, https://arxiv.org/abs/2502.03513; zbMATH Zbl 902660265, https://zbmath.org/902660265
2. Hector Pasten, "The largest prime factor of $n^{2}+1$ and improvements on subexponential $ABC$", Inventiones mathematicae 236 (2024), 373–385. DOI 10.1007/s00222-024-01244-6, https://doi.org/10.1007/s00222-024-01244-6
3. Henryk Iwaniec, "Almost-primes represented by quadratic polynomials", Inventiones Mathematicae 47 (1978), 171–188. DOI 10.1007/BF01578070, https://doi.org/10.1007/BF01578070
4. János Pintz, "Landau's problems on primes", Journal de théorie des nombres de Bordeaux 21 (2009), 357–404. DOI 10.5802/jtnb.676, https://doi.org/10.5802/jtnb.676
5. Paul T. Bateman and Roger A. Horn, "A heuristic asymptotic formula concerning the distribution of prime numbers", Mathematics of Computation 16 (1962), 363–367. DOI 10.1090/s0025-5718-1962-0148632-7, https://doi.org/10.1090/s0025-5718-1962-0148632-7
6. Andrzej Schinzel and Wacław Sierpiński, "Sur certaines hypothèses concernant les nombres premiers", Acta Arithmetica 4 (1958), 185–208. DOI 10.4064/aa-4-3-185-208, https://doi.org/10.4064/aa-4-3-185-208
7. G. H. Hardy and J. E. Littlewood, "Some problems of `Partitio numerorum'; III: On the expression of a number as a sum of primes", Acta Mathematica 44 (1923), 1–70. DOI 10.1007/BF02403921, https://doi.org/10.1007/BF02403921
8. Laurent Hodges, "A Lesser-Known Goldbach Conjecture", Mathematics Magazine 66 (1993), 45–47. DOI 10.1080/0025570X.1993.11996077, https://doi.org/10.1080/0025570X.1993.11996077
9. Marek Wolf, "Some conjectures on primes of the form $m^2+1$", Journal of Combinatorics and Number Theory 5 (2013), 103–131. zbMATH Zbl 06342084, https://zbmath.org/6342084; preprint version: Marek Wolf, "Search for primes of the form $m^2+1$", arXiv:0803.1456, https://arxiv.org/abs/0803.1456
10. OEIS Foundation, "A083844: Number of primes of the form $x^2+1<10^n$", On-Line Encyclopedia of Integer Sequences. https://oeis.org/A083844
11. Lingyu Guo and Victor Zhenyu Guo, "Piatetski-Shapiro Primes in short intervals", preprint, 2026. arXiv:2606.01115, https://arxiv.org/abs/2606.01115