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

1,1

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.

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A089489.

Sequence in context: A245822 A357260 A069797 * A233249 A330573 A309369

Adjacent sequences: A158976 A158977 A158978 * A158980 A158981 A158982

Keyword

easy,nonn

Author

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

Extensions

Edited and entries verified by Klaus Brockhaus, Apr 12 2009

Corrected by Harvey P. Dale, Aug 09 2023

Status

approved