Primality proof for n = 5288447750321988791615322464262168318627237463714249754277190395559387193645633287411691377616107457765456896611395456028803:

Take b = 2.

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

104033754239631823712255190956345641806807791724720405509948501077933969978245219141810136352471050222510526161 is prime.
b^((n-1)/104033754239631823712255190956345641806807791724720405509948501077933969978245219141810136352471050222510526161)-1 mod n = 1993871310070017454814594018773014304438108007971959778916211383450853530933417105655335753894335434212726363576046438925706, which is a unit, inverse 1101059983878053530959011349252208952924383090930639119712903376437540251309407372550758779559025007258243319007801336488032.

(104033754239631823712255190956345641806807791724720405509948501077933969978245219141810136352471050222510526161) divides n-1.

(104033754239631823712255190956345641806807791724720405509948501077933969978245219141810136352471050222510526161)^2 > n.

n is prime by Pocklington's theorem.