38685...97803 (26-digits)

This number is a prime.

                                                386856 2622766813 3590597803

n^2 + 2n + 1 is always a square, but 2^n + 2n + 1 can be prime (starting with n = 0, 1, 5, 13, 65). 2^85 + 2*85 + 1 is the largest such prime for n < 200. [Post]