Primality proof for n = 312289:
<p>Take b = 2.
<p>b^(n-1) mod n = 1.
<p><a href=3253.html>3253 is prime.</a>
<br>b^((n-1)/3253)-1 mod n = 257256, which is a unit, inverse 24151.
<p>(3253) divides n-1.
<p>(3253)^2 > n.
<p>n is prime by Pocklington's theorem.