A160024 Primes p such that p^4 + 11^4 + 3^4 is prime.

7, 11, 13, 19, 23, 31, 41, 47, 61, 67, 73, 83, 101, 107, 127, 157, 163, 191, 193, 277, 281, 311, 337, 373, 379, 401, 409, 431, 443, 461, 491, 523, 541, 569, 607, 643, 673, 691, 719, 733, 743, 757, 769, 887, 929, 947, 953, 1031, 1039, 1087, 1093, 1097, 1103, 1109

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 = 11, r = 3.

It is conjectured that the sequence is infinite.

There are prime twins (11, 13) and other consecutive primes (7, 11; 1093, 1097) in the sequence.

Links

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

Example

p = 3: 3^4 + 11^4 + 3^4 = 14803 = 113*131, so 3 is not in the sequence.
p = 7: 7^4 + 11^4 + 3^4 = 17123 is prime, so 7 is in the sequence.
p = 11: 11^4 + 11^4 + 3^4 = 29363 is prime, so 11 is in the sequence.
p = 13: 13^4 + 11^4 + 3^4 = 43283 is prime, so 13 is in the sequence.

Mathematica

Select[Prime[Range[200]], PrimeQ[#^4+14722]&] (* Harvey P. Dale, Apr 18 2023 *)

Prog

(Magma) [ p: p in PrimesUpTo(1110) | IsPrime(p^4+14722) ]; // Klaus Brockhaus, May 03 2009

Crossrefs

Cf. A158979, A159829, A160022.

Sequence in context: A067466 A091932 A165349 * A063911 A087489 A155488

Adjacent sequences: A160021 A160022 A160023 * A160025 A160026 A160027

Keyword

easy,nonn

Author

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

Extensions

Edited and extended beyond 461 by Klaus Brockhaus, May 03 2009

Status

approved