Primality proof for n = 3086977169:
Take b = 2.
b^(n-1) mod n = 1.
19211 is prime.
b^((n-1)/19211)-1 mod n = 2446132186, which is a unit, inverse 1684996232.
83 is prime.
b^((n-1)/83)-1 mod n = 1975856084, which is a unit, inverse 1992305034.
(83 * 19211) divides n-1.
(83 * 19211)^2 > n.
n is prime by Pocklington's theorem.