Primality proof for n = 31757755568855353:

Take b = 2.

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

430751 is prime.
b^((n-1)/430751)-1 mod n = 15280470251845349, which is a unit, inverse 15419847897348699.

4153 is prime.
b^((n-1)/4153)-1 mod n = 28256283979054929, which is a unit, inverse 25117532289192573.

(4153 * 430751) divides n-1.

(4153 * 430751)^2 > n.

n is prime by Pocklington's theorem.