A079544 Primes of the form x^2 + y^2 + 1, x>0, y>0.

3, 11, 19, 41, 53, 59, 73, 83, 101, 107, 131, 137, 149, 163, 179, 181, 227, 233, 251, 293, 307, 347, 389, 401, 443, 467, 491, 521, 523, 563, 587, 593, 613, 641, 677, 739, 773, 809, 811, 821, 883

Offset

1,1

Comments

Sequence is known to be infinite due to a result of Linnik.

References

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 11.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Aminu Alhaji Ibrahim, Sa’idu Isah Abubaka, Aunu Integer Sequence as Non-Associative Structure and Their Graph Theoretic Properties, Advances in Pure Mathematics, 2016, 6, 409-419.

Ju. V. Linnik. An asymptotic formula in an additive problem of Hardy and Littlewood (Russian). Izv. Akad. Nauk SSSR, ser. math., 24:629-706, 1960. Cited in Matomäki 2007.

Kaisa Matomäki, Prime numbers of the form p = m^2+n^2+1 in short intervals, Acta Arith. 128 (2007), pp. 193-200.

Yu-Chen Sun and Hao Pan, The Green-Tao theorem for primes of the form x^2+y^2+1, arXiv preprint arXiv:1708.08629 [math.NT], 2017.

Mathematica

iMax=7!; a=Floor[Sqrt[iMax]]; lst={}; Do[Do[p=x^2+y^2+1; If[PrimeQ@p&&p<=iMax, AppendTo[lst, p]], {y, 1, a}], {x, 1, a}]; Union[lst] (* Vladimir Joseph Stephan Orlovsky, Aug 11 2009 *)

Prog

(PARI) list(lim)=my(v=List(), t); lim\=1; for(x=1, sqrtint(lim-2), forstep(y=2-x%2, min(x, sqrtint(lim-x^2-1)), 2, if(isprime(t=x^2+y^2+1), listput(v, t)))); vecsort(Vec(v), , 8) \\ Charles R Greathouse IV, Jun 13 2012

Crossrefs

Cf. A079545.

Sequence in context: A138726 A219527 A161429 * A192717 A163183 A007520

Adjacent sequences: A079541 A079542 A079543 * A079545 A079546 A079547

Keyword

nonn

Author

N. J. A. Sloane, Jan 23 2003

Status

approved