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

1,1

Comments

It is conjectured that there are infinitely many primes of the form x^2 + 1 (and thus this sequence never becomes constant), but this has not been proved.

These primes can be found quickly using a sieve based on the fact that numbers of this form have at most one primitive prime factor (A005529). The sum of the reciprocals of these primes is 0.81459657... - T. D. Noe, Oct 14 2003

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

Table of n, a(n) for n=1..28.

C. K. Caldwell, An Amazing Prime Heuristic A pdf file.

Jon Grantham, Parallel Computation of Primes of the form x^2+1

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.

Example

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.

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}]

Crossrefs

Cf. A005574, A002496, A083845, A083846, A083847, A083848, A083849, A214452, A214454, A214455.

Cf. A005529 (primitive prime factors of the sequence k^2+1).

Sequence in context: A219555 A263738 A011963 * A026554 A259132 A369391

Adjacent sequences: A083841 A083842 A083843 * A083845 A083846 A083847

Keyword

nonn

Author

Harry J. Smith, May 05 2003

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

Status

approved