1,1

It is conjectured that this sequence is infinite, but this has never 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 (noe(AT)sspectra.com), Oct 14 2003

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.

C. K. Caldwell, AN AMAZING PRIME HEURISTIC A pdf file.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. Landau's Problems.

Marek Wolf, Search for primes of the form m^2+1

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.

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}]

Cf. A005574, A002496, A083845, A083846, A083847, A083848, A083849.

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

Sequence in context: A079162 A043330 A011963 this_sequence A026554 A099413 A127392

Adjacent sequences: A083841 A083842 A083843 this_sequence A083845 A083846 A083847

nonn

Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 05 2003

Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), May 08 2003

More terms from T. D. Noe (noe(AT)sspectra.com), Oct 14 2003

a(17)-a(22) from Robert Gerbicz (robert.gerbicz(AT)gmail.com), Apr 15 2009