Primality proof for n = 134209075623850670257907127035661234387627594536271696043327:

Take b = 2.

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

32037243671365622938525385539447601257252076559602597 is prime.
b^((n-1)/32037243671365622938525385539447601257252076559602597)-1 mod n = 102875901498147389893355324847111490320978515801439890104110, which is a unit, inverse 5795044602282436819321274109405227888063888110556036704582.

(32037243671365622938525385539447601257252076559602597) divides n-1.

(32037243671365622938525385539447601257252076559602597)^2 > n.

n is prime by Pocklington's theorem.