A160022 Primes p such that p^4 + 5^4 + 3^4 is prime.

3, 23, 47, 53, 67, 73, 89, 101, 103, 109, 151, 157, 179, 229, 521, 557, 569, 619, 661, 821, 977, 1013, 1087, 1129, 1277, 1321, 1451, 1559, 1607, 1627, 1741, 1867, 1871, 1949, 2137, 2389, 2441, 2797, 3271, 3313, 3643, 3677, 3769, 3847, 4001, 4027, 4133

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 = 5, r = 3.

It is conjectured that the sequence is infinite.

There are twin prime (101, 103) and other consecutive primes (151, 157; 1867, 1871) in the sequence.

Examples

			p = 3: 3^4 + 5^4 + 3^4 = 787 is prime, so 3 is in the sequence.
p = 5: 5^4 + 5^4 + 3^4 = 1331 = 11^3, so 5 is not in the sequence.
p = 101: 101^4 + 5^4 + 3^4 = 104061107 is prime, so 101 is in the sequence.
p = 103: 103^4 + 5^4 + 3^4 = 112551587 is prime, so 103 is in the sequence.
		

Crossrefs

Cf. A158979, A159829, A160031.

Programs

Magma

[p: p in PrimesUpTo(5000)|IsPrime(p^4+706)] // Vincenzo Librandi, Dec 18 2010

Mathematica

With[{c=5^4+3^4},Select[Prime[Range[600]],PrimeQ[#^4+c]&]] (* Harvey P. Dale, Aug 14 2011 *)

PARI

is(n)=isprime(n) && isprime(n^4+706) \\ Charles R Greathouse IV, Jun 07 2016

Extensions

Edited, 1607 inserted and extended beyond 3643 by Klaus Brockhaus, May 03 2009