Primality proof for n = 1293908032763388459087425944272364834610609736614628384140627657852601712102489824169817319028562969141747131097192516171:
<p>Take b = 2.
<p>b^(n-1) mod n = 1.
<p><a href=509834927133261305493315894436753153401122015851834224693667.html>509834927133261305493315894436753153401122015851834224693667 is prime.</a>
<br>b^((n-1)/509834927133261305493315894436753153401122015851834224693667)-1 mod n = 978901105006250115884915286145519688671922318990340231872739107023895321773797064555046744862583038093865880012415245956, which is a unit, inverse 250218387412233364138834004728518384375595769921446304721183449157162108270401645837675792472820955327080267897422582491.
<p><a href=1011824009757342167700258385900659940913.html>1011824009757342167700258385900659940913 is prime.</a>
<br>b^((n-1)/1011824009757342167700258385900659940913)-1 mod n = 585743935265560697126900886024400684669547043654761220537676480076739780887832782485471512474991110321639885524127561753, which is a unit, inverse 974127321904902165780159792779393027983424113056022246973478600291985529847116851958345289804981494420751872968934518657.
<p>(1011824009757342167700258385900659940913 * 509834927133261305493315894436753153401122015851834224693667) divides n-1.
<p>(1011824009757342167700258385900659940913 * 509834927133261305493315894436753153401122015851834224693667)^2 > n.
<p>n is prime by Pocklington's theorem.