Primality proof for n = 31401709:
<p>Take b = 2.
<p>b^(n-1) mod n = 1.
<p><a href=1993.html>1993 is prime.</a>
<br>b^((n-1)/1993)-1 mod n = 12526448, which is a unit, inverse 1373451.
<p><a href=101.html>101 is prime.</a>
<br>b^((n-1)/101)-1 mod n = 584585, which is a unit, inverse 13470207.
<p>(101 * 1993) divides n-1.
<p>(101 * 1993)^2 > n.
<p>n is prime by Pocklington's theorem.