Primality proof for n = 9511259360250244436150360929400467:

Take b = 2.

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

7057776167979264311 is prime.
b^((n-1)/7057776167979264311)-1 mod n = 4410575249180774820092423681056164, which is a unit, inverse 9131088010384841444278586403721644.

(7057776167979264311) divides n-1.

(7057776167979264311)^2 > n.

n is prime by Pocklington's theorem.