Primality proof for n = 5288447750321988791615322464262168318627237463714249754277190395559387193645633287411691377616107457765456896611395456028803:
<p>Take b = 2.
<p>b^(n-1) mod n = 1.
<p><a href=104033754239631823712255190956345641806807791724720405509948501077933969978245219141810136352471050222510526161.html>104033754239631823712255190956345641806807791724720405509948501077933969978245219141810136352471050222510526161 is prime.</a>
<br>b^((n-1)/104033754239631823712255190956345641806807791724720405509948501077933969978245219141810136352471050222510526161)-1 mod n = 1993871310070017454814594018773014304438108007971959778916211383450853530933417105655335753894335434212726363576046438925706, which is a unit, inverse 1101059983878053530959011349252208952924383090930639119712903376437540251309407372550758779559025007258243319007801336488032.
<p>(104033754239631823712255190956345641806807791724720405509948501077933969978245219141810136352471050222510526161) divides n-1.
<p>(104033754239631823712255190956345641806807791724720405509948501077933969978245219141810136352471050222510526161)^2 > n.
<p>n is prime by Pocklington's theorem.