A126769 Primes p of the form k^4 + s where k > 1, s >= 1 and k^2 + s is also prime.

17, 19, 23, 29, 31, 41, 43, 53, 59, 71, 73, 79, 83, 89, 101, 103, 109, 113, 131, 139, 149, 151, 163, 173, 179, 181, 191, 193, 199, 211, 223, 229, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 337, 347, 349, 353, 359, 367, 379, 383, 389

Offset: 1

Author

Tomas Xordan, Feb 16 2007

Keywords

• nonn

Comments

For primes not in this sequence see A128292.

Conjecture: Every prime q > 3 can be written in a nontrivial way as the sum of two or more squares, q = Sum_{i} (k_i)^2, such that the sum of the fourth powers of the squared numbers is again prime, p = Sum_{i} (k_i)^4. (Tomas Xordan)

This sequence illustrates an easy case of the conjecture: For primes q arising in the sequence there exists an integer k > 1, a positive integer s and a prime p such that k^2 < q, s = q - k^2, p = k^4 + s and p > q.

This corresponds to the case where only one of the squares is larger than 1 and all other s terms are equal to 1. - M. F. Hasler, May 23 2018

Examples

			19 = 2^4+3 is prime and 2^2+3 = 7 is a smaller prime, hence 19 is a term.
23 = 2^4+7 is prime and 2^2+7 = 11 is a smaller prime, hence 23 is a term.
1307 = 6^4+11 is prime and 6^2+11 = 47 is a smaller prime, hence 1307 is a term.
37 is prime, 2^4+21 is the only way to write 37 as k^4+s, but neither 2^2+21 = 25 nor 3^2+21 = 30 are prime, hence 37 is not in the sequence.
		

Links

• R. J. Cano, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A128292.

Programs

PARI

{m=5;v=[];for(n=2,m,for(k=1,(m+1)^4,if(isprime(p=n^4+k)&&pKlaus Brockhaus, Feb 24 2007

Extensions

Edited, corrected and extended by Klaus Brockhaus, Feb 24 2007

Name edited following a suggestion from R. Sigrist, and the conjecture rephrased by M. F. Hasler, May 23 2018