A121411 Positive integers k for which there are primes of the form a^2+k^n=b^2+k^m with positive integers (a,b,m,n) and a > b.

2, 5, 6, 8, 10, 12, 13, 17, 18, 20, 21, 22, 26, 28, 30, 32, 33, 37, 38, 40, 42, 45, 46, 48, 50, 52, 53, 56, 58, 60, 61, 62, 65, 66, 68, 70, 72, 76, 77, 78, 80, 82, 85, 86, 88, 90, 92, 93, 96, 97, 98

Offset

1,1

Comments

The sequence is "hard" in the sense that it not known how to prove that the necessary conditions are sufficient for the existence of primes.

Links

Table of n, a(n) for n=1..51.

David Broadhurst and Mike Oakes, Primes of the form a^2 + k^n = b^2 + k^m.

David Broadhurst and Mike Oakes, proof of the necessity the conditions given for the conjectured generating method.

Formula

Conjecturally, a(n) is the n-th positive nonsquare integer that is not congruent to -1 mod 4, nor to -1 mod 5, nor to -7 mod 16.

Example

a(5455)=9998 because it was possible to find primes of the form a^2 + k^n = b^2 + k^m with positive integers (a,b,k,m,n), a > b, k < 10^4 and k satisfying the proved necessary conditions of the conjectured generating method.

Prog

(PARI) {ls=[]; for(k=1, 10^4, if(!issquare(k)&&(k+1)%4&&(k+1)%5&&(k+7)%16, ls=concat(ls, k))); print(ls)}

Crossrefs

Sequence in context: A087943 A034020 A187476 * A224889 A363676 A047441

Adjacent sequences: A121408 A121409 A121410 * A121412 A121413 A121414

Keyword

hard,nonn

Author

David Broadhurst, Jul 29 2006

Status

approved