Primality proof for n = 837987995621412318723376562387865382967460363787024586107722590232610251879607410804876779383055508762141059258497448934987052508775626162460930737942299:
<p>Take b = 2.
<p>b^(n-1) mod n = 1.
<p><a href=21682369618568459986649847415957985919370700162617.html>21682369618568459986649847415957985919370700162617 is prime.</a>
<br>b^((n-1)/21682369618568459986649847415957985919370700162617)-1 mod n = 254286029045568053158871710695275304591527992141003398255941978576982727347119351876714225346994080492196302854051973037057220734040558096920682009131375, which is a unit, inverse 640381316677242288578606172565549962768756501448352981159227620280360516710912921879949621517217641882210482616372105698114492378403165810180440258810398.
<p><a href=27727598583240555671506730615016813004130177.html>27727598583240555671506730615016813004130177 is prime.</a>
<br>b^((n-1)/27727598583240555671506730615016813004130177)-1 mod n = 458179180417942970616723611082879192922827379278467822980383100971456326849935941131321025225214884682372240695739139766766598629539734545065810606041123, which is a unit, inverse 381027914352246182540083285518147125423708937528805457938086517508288145169850711498293270941986505359764342984595567253669635271475503067084726994281213.
<p>(27727598583240555671506730615016813004130177 * 21682369618568459986649847415957985919370700162617) divides n-1.
<p>(27727598583240555671506730615016813004130177 * 21682369618568459986649847415957985919370700162617)^2 > n.
<p>n is prime by Pocklington's theorem.