Number of pairs of numbers (n,x) with 0 < x <= n such that n^2 + x^2 is prime.

1, 1, 1, 1, 2, 2, 1, 3, 1, 4, 2, 1, 4, 3, 3, 3, 4, 3, 4, 6, 2, 4, 5, 3, 7, 6, 4, 4, 4, 4, 7, 6, 5, 6, 8, 5, 6, 7, 3, 9, 5, 5, 8, 8, 7, 9, 6, 7, 10, 8, 6, 9, 10, 5, 8, 8, 6, 10, 11, 8, 11, 10, 6, 9, 15, 5, 10, 11, 4, 11, 13, 6, 12, 10, 12, 11, 9, 8, 11, 19, 10

1

It appears that for every n, there is an x in the range [0,n] such that n^2 + x^2 is prime.

T. D. Noe, Plot of 5000 terms

T. D. Noe, Table of 5000 terms

(Mma) Table[cnt = 0; Do[If[PrimeQ[x^2 + y^2], cnt++], {y, x}]; cnt, {x, 100}]

nonn

T. D. Noe, Apr 24 2015