A158979 a(n) is the smallest number > n such that n^4 + a(n)^4 is prime.

2, 3, 4, 5, 8, 7, 10, 9, 10, 13, 16, 13, 14, 15, 22, 17, 20, 23, 24, 29, 38, 29, 26, 41, 26, 27, 28, 33, 34, 37, 32, 37, 34, 35, 52, 37, 38, 39, 46, 41, 50, 53, 44, 47, 58, 55, 50, 49, 60, 61, 62, 61, 56, 55, 58, 59, 68, 61, 62, 73, 66, 77, 64, 67, 84, 71

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 01 2009

Keywords

• easy
• nonn

Comments

For exponent 2 instead of 4 see A089489: Pythagorean triple has a prime hypotenuse.

Corresponding sequences with odd exponent u are impossible: x^u + y^u has factor x+y.

a(2k-1) is even, a(2k) is odd, a(n)-n is odd.

Conjecture: a(n) exists for all n, i.e., the sequence is well-defined and infinite.

Conjecture: a(n)-n = 1 for infinitely many n.

The largest value of a(n)-n for n <= 100 occurs at n = 90: 121-90 = 31.

a(n)-n = 1 for 35 values of n <= 100.

Examples

			1^4 + 2^4 = 17 is prime, so a(1) = 2.
2^4 + 3^4 = 97 is prime, so a(2) = 3.
5^4 + 6^4 = 1921 = 17*113, 5^4 + 7^4 = 3026 = 2*17*89, 5^4 + 8^4 = 4721 is prime, so a(5) = 8.
		

Links

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

Crossrefs

Cf. A089489.

Programs

Magma

S:=[]; for n in [1..72] do q:=n^4; k:=n+1; while not IsPrime(q+k^4) do k+:=1; end while; Append(~S, k); end for; S; // Klaus Brockhaus, Apr 12 2009

Mathematica

sn[n_]:=Module[{k=n+1,n4=n^4},While[CompositeQ[n4+k^4],k++];k]; Array[sn,80] (* Harvey P. Dale, Aug 09 2023 *)

Extensions

Edited and entries verified by Klaus Brockhaus, Apr 12 2009

Corrected by Harvey P. Dale, Aug 09 2023