A160025 Primes p such that p^4 + 13^4 + 3^4 is prime.

3, 11, 13, 17, 31, 41, 43, 53, 83, 127, 167, 181, 193, 211, 241, 311, 337, 349, 421, 431, 487, 521, 557, 613, 617, 647, 701, 769, 811, 857, 953, 1021, 1151, 1249, 1289, 1303, 1373, 1453, 1459, 1471, 1523, 1553, 1567, 1579, 1613, 1663, 1669, 1747, 1823, 1831

Offset

1,1

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 = 13, r = 3.

It is conjectured that the sequence is infinite.

There are prime twins (11, 13) and other consecutive primes (421, 431; 1823, 1831) in the sequence.

Links

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

Example

p = 3: 3^4 + 13^4 + 3^4 = 28723 is prime, so 3 is in the sequence.
p = 5: 5^4 + 13^4 + 3^4 = 29267 = 7*37*113, so 5 is not in the sequence.
p = 17: 17^4 + 13^4 + 3^4 = 112163 is prime, so 17 is in the sequence.
p = 83: 83^4 + 13^4 + 3^4 = 47486963 is prime, so 83 is in the sequence.

Mathematica

Select[Prime[Range[400]], PrimeQ[#^4+28642]&] (* Harvey P. Dale, Dec 14 2011 *)

Prog

(Magma) [ p: p in PrimesUpTo(1840) | IsPrime(p^4+28642) ]; // Klaus Brockhaus, May 03 2009

Crossrefs

Cf. A158979, A159829, A160022.

Sequence in context: A020614 A191026 A158296 * A045427 A115165 A050583

Adjacent sequences: A160022 A160023 A160024 * A160026 A160027 A160028

Keyword

easy,nonn

Author

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

Extensions

Edited and extended beyond 857 by Klaus Brockhaus, May 03 2009

Status

approved