A083844 Number of primes of the form x^2 + 1 < 10^n.
2, 4, 10, 19, 51, 112, 316, 841, 2378, 6656, 18822, 54110, 156081, 456362, 1339875, 3954181, 11726896, 34900213, 104248948, 312357934, 938457801, 2826683630, 8533327397, 25814570672, 78239402726, 237542444180, 722354138859, 2199894223892
Offset: 1
Keywords
Examples
a(3) = 10 because the only primes or the form x^2 + 1 < 10^3 are the ten primes: 2, 5, 17, 37, 101, 197, 257, 401, 577, 677.
References
- G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.
- P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 190.
Links
- C. K. Caldwell, An Amazing Prime Heuristic A pdf file.
- Jon Grantham, Parallel Computation of Primes of the form x^2+1
- Jon Grantham and Hester Graves, Primes of the Form m^2+1 and Goldbach's 'Other Other' Conjecture, arXiv:2502.03513 [math.NT], 2025.
- Apoloniusz Tyszka, Sławomir Kurpaska, Open problems that concern computable sets X, subset of N, and cannot be formally stated as they refer to current knowledge about X, (2020).
- Eric Weisstein's World of Mathematics, Landau's Problems.
- Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.
Crossrefs
Programs
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Mathematica
c = 1; k = 2; (* except for the initial prime 2, all X's must be odd. *) Do[ While[ k^2 + 1 < 10^n, If[ PrimeQ[k^2 + 1], c++ ]; k += 2]; Print[c], {n, 1, 20}]
Extensions
Edited by Robert G. Wilson v, May 08 2003
More terms from T. D. Noe, Oct 14 2003
a(17)-a(22) from Robert Gerbicz, Apr 15 2009
a(23)-a(25) from Marek Wolf and Robert Gerbicz (code from Robert, computation done by Marek) Robert Gerbicz, Mar 13 2010
a(26)-a(28) from Jon Grantham, Jan 18 2017
a(28) corrected by Jon Grantham, Jan 30 2018
Comments