A160031 Primes p such that p^4 + 2*3^4 is prime.

5, 13, 19, 43, 71, 83, 97, 101, 107, 109, 127, 149, 179, 193, 197, 211, 233, 241, 311, 353, 383, 401, 421, 541, 577, 599, 607, 619, 641, 647, 683, 709, 727, 751, 769, 827, 877, 883, 941, 967, 991, 1009, 1061, 1097, 1109, 1187, 1289, 1373, 1381, 1409, 1439

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 30 2009

Keywords

• easy
• nonn

Comments

For primes p, q, r the sum p^4 + q^4 + r^4 can be prime only if at least one of p, q, r equals 3. This sequence is the special case q = r = 3.

It is conjectured that the sequence is infinite.

There are prime twins (107, 109) and other consecutive primes (193, 197) in the sequence.

Examples

			p = 5: 5^4 + 2*3^4 = 787 is prime, so 5 is in the sequence.
p = 7: 7^4 + 2*3^4 = 2563 = 11*233, so 7 is not in the sequence.
p = 107: 107^4 + 2*3^4 = 131079763 is prime, so 107 is in the sequence.
p = 109: 109^4 + 2*3^4 = 141158323 is prime, so 109 is in the sequence.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A158979, A159829, A160022.

Programs

Magma

[ p: p in PrimesUpTo(1450) | IsPrime(p^4+162) ]; // Klaus Brockhaus, May 03 2009

Mathematica

Select[Prime[Range[300]],PrimeQ[#^4+162]&] (* Harvey P. Dale, May 10 2018 *)

PARI

is(n)=isprime(n) && isprime(n^4+162) \\ Charles R Greathouse IV, Jun 07 2016

Extensions

Edited and extended beyond 683 by Klaus Brockhaus, May 03 2009