Primality proof for n = 3376384040438327312767543585515740988479041826522993:

Take b = 2.

b^(n-1) mod n = 1.

592299861982871449195634528262215319398393153 is prime.
b^((n-1)/592299861982871449195634528262215319398393153)-1 mod n = 325033923240904834364088111477035952828626849903838, which is a unit, inverse 1655108222118585084429848496419637768180570536721849.

(592299861982871449195634528262215319398393153) divides n-1.

(592299861982871449195634528262215319398393153)^2 > n.

n is prime by Pocklington's theorem.