Primality proof for n = 252243041490141992091916747959634815169496193136845853336769661422832941962389811668187387649516185742186371754593:
<p>Take b = 2.
<p>b^(n-1) mod n = 1.
<p><a href=38916407753682983861283783766663832991809018999410768212336775108672551624633480899.html>38916407753682983861283783766663832991809018999410768212336775108672551624633480899 is prime.</a>
<br>b^((n-1)/38916407753682983861283783766663832991809018999410768212336775108672551624633480899)-1 mod n = 912365614689170617086543404626706406500731224071477834893272335532724899852886083602929504564654840016389225160, which is a unit, inverse 214916641123192098059566615049466610744613774242490803148394649095293696073679794688712840892528844785633430366111.
<p>(38916407753682983861283783766663832991809018999410768212336775108672551624633480899) divides n-1.
<p>(38916407753682983861283783766663832991809018999410768212336775108672551624633480899)^2 > n.
<p>n is prime by Pocklington's theorem.