No, it can never be prime

If N is even, then it is divisible by 3 (prove by considering the equation mod 3)

If N = 2m + 1, then it is divisible by 13 (prove by considering the equation mod 13). The use of 13 was indicated directly by noting that 19*8 + 17 = 169, whose only prime factor is 13.

If N = 2m + 3, then it is divisible by 5 (prove by considering the equation mod 5).