A Sierpiński number of the first kind is a number of the form S_n congruent n^n + 1. The first few are 2, 5, 28, 257, 3126, 46657, 823544, 16777217, ... (OEIS A014566). Sierpiński proved that if S_n is prime with n>=2, then n must be of the form n = 2^(2^k), making S_n a Fermat number F_m with m = k + 2^k. The first few m of this form are 1, 3, 6, 11, 20, 37, 70, ... (OEIS A006127). The numbers of digits in the number S_k is given by d_k = ⌈2^(k + 2^k) log_10 2⌉, where ⌈z⌉ is the ceiling function, so the numbers of digits in the first few candidates are 1, 3, 20, 617, 315653, 41373247568, ... (OEIS A089943).

Cullen number | Cunningham number | Fermat number | Sierpiński number of the second kind | Woodall number

Wacław Sierpiński