Primality proof for n = 393761:
Take b = 2.
b^(n-1) mod n = 1.
107 is prime.
b^((n-1)/107)-1 mod n = 275749, which is a unit, inverse 230230.
23 is prime.
b^((n-1)/23)-1 mod n = 267571, which is a unit, inverse 186914.
(23 * 107) divides n-1.
(23 * 107)^2 > n.
n is prime by Pocklington's theorem.