Primality proof for n = 82357043705861571042799942460715218111:
<p>Take b = 2.
<p>b^(n-1) mod n = 1.
<p><a href=1176529195798022443468570606581645973.html>1176529195798022443468570606581645973 is prime.</a>
<br>b^((n-1)/1176529195798022443468570606581645973)-1 mod n = 1180591620717411303423, which is a unit, inverse 28400898209859348889760089546502274558.
<p>(1176529195798022443468570606581645973) divides n-1.
<p>(1176529195798022443468570606581645973)^2 > n.
<p>n is prime by Pocklington's theorem.