Primes of the Form m2+1 and Goldbach’s `Other Other’ Conjecture
Grantham Jon 1, Graves Hester 2
Institute for Defense Analyses
Center for Computing Sciences
17100 Science Drive
Bowie, Maryland 20715 USA
1 E-mail: grantham@super.org
2 E-mail: hkgrave@super.org
Received:
Received: 9 March 2025; revised: 23 July 2025; accepted: 24 July 2025; published online: 7 August 2025
DOI: 10.12921/cmst.2025.0000004
Abstract:
We compute all primes up to 6.25 × 1028 of the form m2 + 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.
Key words:
Bateman-Horn, Goldbach, Hypothesis H, primes, sieve of Eratosthenes, sums of squares
References:
[1] P.T. Bateman, R.A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16, 363–367 (1962).
[2] R. Crandall, C. Pomerance, Prime Numbers: A Computational Perspective, Springer, New York, second edition (2005).
[3] Project Euler, Goldbach’s other conjecture, https://projecteuler.net/problem=46.
[4] J. Friedlander, H. Iwaniec, The polynomial X2 +Y 4 captures its primes, Ann. Math. (2) 148, 945–1040 (1998).
[5] P.-H. Fuss, Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle. Tomes I, II, Précédée d’une notice sur les travaux de Léonard Euler, tant imprimés qu’inédits et publiée sous les auspices de l’Académie Impériale des Sciences de Saint-Pétersbourg. The Sources of Science, No. 35, Johnson Reprint Corp., New York-London (1968).
[6] G.H. Hardy, J.E. Littlewood, Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes, Acta Math. 44, 1–70 (1923).
[7] L. Hodges, A Lesser-Known Goldbach Conjecture, Math. Mag. 66, 45–47 (1993).
[8] H. Pasten, The largest prime factor of n2 + 1 and improvements on subexponential ABC, Invent. Math. 236, 373–385 (2024).
[9] J. Pintz, Landau’s problems on primes, J. Théor. Nombres Bordeaux 21, 57–404 (2009).
[10] A. Schinzel, W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arith. 4, 185–208 (1958); erratum 5, 259 (1958).
[11] M. Wolf, R. Gerbicz, The On-line Encyclopedia of Integer Sequences (2010), https://oeis.org/A083844.
[12] M. Wolf, Some conjectures on primes of the form m2+1 Comb. Number Theory 5, 103–131 (2013).
We compute all primes up to 6.25 × 1028 of the form m2 + 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.
Key words:
Bateman-Horn, Goldbach, Hypothesis H, primes, sieve of Eratosthenes, sums of squares
References:
[1] P.T. Bateman, R.A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16, 363–367 (1962).
[2] R. Crandall, C. Pomerance, Prime Numbers: A Computational Perspective, Springer, New York, second edition (2005).
[3] Project Euler, Goldbach’s other conjecture, https://projecteuler.net/problem=46.
[4] J. Friedlander, H. Iwaniec, The polynomial X2 +Y 4 captures its primes, Ann. Math. (2) 148, 945–1040 (1998).
[5] P.-H. Fuss, Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle. Tomes I, II, Précédée d’une notice sur les travaux de Léonard Euler, tant imprimés qu’inédits et publiée sous les auspices de l’Académie Impériale des Sciences de Saint-Pétersbourg. The Sources of Science, No. 35, Johnson Reprint Corp., New York-London (1968).
[6] G.H. Hardy, J.E. Littlewood, Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes, Acta Math. 44, 1–70 (1923).
[7] L. Hodges, A Lesser-Known Goldbach Conjecture, Math. Mag. 66, 45–47 (1993).
[8] H. Pasten, The largest prime factor of n2 + 1 and improvements on subexponential ABC, Invent. Math. 236, 373–385 (2024).
[9] J. Pintz, Landau’s problems on primes, J. Théor. Nombres Bordeaux 21, 57–404 (2009).
[10] A. Schinzel, W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arith. 4, 185–208 (1958); erratum 5, 259 (1958).
[11] M. Wolf, R. Gerbicz, The On-line Encyclopedia of Integer Sequences (2010), https://oeis.org/A083844.
[12] M. Wolf, Some conjectures on primes of the form m2+1 Comb. Number Theory 5, 103–131 (2013).
