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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 28 2012
STATUS
approved