A158477 Primes p with property that Q(p) = p^32+2^32 is prime.

29, 59, 101, 103, 109, 181, 199, 379, 769, 881, 919, 977, 1097, 1213, 1303, 1583, 2099, 2113, 2441, 2521, 2617, 2777, 3067, 3739, 4133, 4289, 4519, 4931, 5039, 5113, 5227, 5417, 5743, 5783, 6143, 6373, 6691, 8053, 8209, 8287, 8513, 9109, 9203, 9689, 9787, 9923, 9941

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 20 2009

Keywords

• nonn

Comments

1) Q=(p^16)^2+(2^16)^2 only for Q=4k+1 because of Fermat/Euler/Lagrange theorem concerning prime as sum of two squares.

2) It is conjectured that sequence a(n) is infinite.

3) Note the twin prime: a(3)=101, a(4)=103.

The next set of twins is a(101)=30557, a(102)=30559. - Robert Israel, Apr 05 2016

Examples

			p=3: 3^32+2^32=1853024483819137 = 1153 x 1607133116929 no prime;
also for following primes p=5, 7, 11, 13, 17, 19, 23: Q(p) no prime;
p=29: 29^32+2^32=62623297589448778360828428329074752313100292737 is prime => a(1)=29.
		

References

• Richard E. Crandall, Carl Pomerance, Prime Numbers: A Computational Perspective, Springer 2001.
• Leonard E. Dickson, History of the Theory of Numbers, Dover Pub. Inc., 2005.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A062324, A157764, A157950, A158361.

Programs

Maple

select(t -> isprime(t) and isprime(t^32 + 2^32), [seq(i,i=3..10000,2)]); # Robert Israel, Apr 05 2016

Mathematica

With[{c=2^32},Select[Prime[Range[1300]],PrimeQ[#^32+c]&]] (* Harvey P. Dale, May 04 2018 *)

PARI

isA158477(n) = isprime(n) && isprime(n^32+4294967296) \\ Michael B. Porter, Dec 17 2009

PARI

lista(nn) = forprime(p=3, nn, if(ispseudoprime(p^32+2^32), print1(p, ", "))); \\ Altug Alkan, Apr 05 2016

Formula

n^32+2^32 and n to be prime.