Primality proof for n = 101148471075752777:

Take b = 2.

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

1176035613847 is prime.
b^((n-1)/1176035613847)-1 mod n = 2655626225930651, which is a unit, inverse 48147718573863262.

(1176035613847) divides n-1.

(1176035613847)^2 > n.

n is prime by Pocklington's theorem.