Primality proof for n = 7237005577332262213973186563042994240857116359379907606001950938285454250989:

Take b = 2.

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

276602624281642239937218680557139826668747 is prime.
b^((n-1)/276602624281642239937218680557139826668747)-1 mod n = 2975531125133123119648879457563281269120703404158613135195788908093573672640, which is a unit, inverse 2233364593245673852815932892735013076939656528578070433728021143254617821614.

(276602624281642239937218680557139826668747) divides n-1.

(276602624281642239937218680557139826668747)^2 > n.

n is prime by Pocklington's theorem.