A219791 Number of ways to write n=x+y (0<x<=y) with (xy)^2+1 prime.

0, 1, 1, 1, 2, 0, 2, 1, 2, 2, 2, 2, 2, 2, 5, 0, 2, 1, 2, 2, 4, 2, 4, 0, 6, 2, 6, 2, 5, 3, 6, 3, 5, 4, 7, 3, 6, 2, 5, 6, 6, 1, 6, 5, 4, 1, 6, 2, 7, 5, 5, 2, 9, 3, 8, 4, 8, 3, 6, 6, 4, 3, 9, 4, 13, 4, 9, 4, 5, 9, 2, 1, 11, 4, 14, 4, 10, 3, 9, 8, 4, 3, 6, 5, 10, 3

Offset

1,5

Comments

Conjecture: a(n)>0 if n is different from 1, 6, 16, 24.

This conjecture has been verified for n up to 10^7. It implies that there are infinitely many primes of the form x^2+1.

Zhi-Wei Sun also made the following general conjecture: For any positive integer k, each sufficiently large integer n cna be written as x+y (x>0, y>0) with (xy)^{2^k}+1 prime.

For example, for k=2,3,4 it suffices to require that n is greater than 22, 386, 748 respectively.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, An amazing conjecture on primes, a message to Number Theory List, Nov. 27, 2012.

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Example

a(8)=1 since 8=4+4 with (4*4)^2+1=257 prime.
a(9)=2 since 9=2+7=4+5, and (2*7)^2+1=197 and (4*5)^2+1=401 are prime.

Mathematica

a[n_] := a[n] = Sum[If[PrimeQ[(k(n-k))^2+1] == True, 1, 0], {k, n/2}]; Do[Print[n, " ", a[n]], {n, 100}]

Crossrefs

Cf. A091182, A219782.

Sequence in context: A301564 A334361 A360568 * A029345 A153149 A249072

Adjacent sequences: A219788 A219789 A219790 * A219792 A219793 A219794

Keyword

nonn

Author

Zhi-Wei Sun, Nov 28 2012

Status

approved