Prime values of m^2 - 2*n^2 for prime m and n.
7, 17, 23, 31, 41, 47, 71, 103, 113, 151, 167, 191, 239, 263, 271, 281, 311, 353, 359, 383, 431, 479, 503, 521, 599, 647, 719, 743, 823, 839, 863, 887, 911, 953, 983, 1031, 1103, 1151, 1223, 1319, 1361, 1367, 1439, 1487, 1511, 1559, 1583, 1607, 1663, 1823
1
Granville gives this as an example of a sequence probably containing an infinite number of primes.
T. D. Noe, Plot of 1000 terms
T. D. Noe, Table of 1000 terms
Andrew Granville, Primes in intervals of bounded length, arXiv 1410.8400 (Oct 30 2014)
(Mma) nn = 100; Select[Union[Flatten[Table[m^2 - 2 n^2, {m, Prime[Range[nn]]}, {n, Prime[Range[nn]]}]]], # > 0 && # < nn^2 && PrimeQ[#] &]
nonn
T. D. Noe, Nov 04 2014