Primality proof for n = 17942476279071857634847638457723:

Take b = 2.

b^(n-1) mod n = 1.

2089326636302777 is prime.
b^((n-1)/2089326636302777)-1 mod n = 8888586795800304819756846599628, which is a unit, inverse 12531956664956805667278672456760.

1300347341 is prime.
b^((n-1)/1300347341)-1 mod n = 3012345522519017498484093476900, which is a unit, inverse 9263424345051224582796052959711.

(1300347341 * 2089326636302777) divides n-1.

(1300347341 * 2089326636302777)^2 > n.

n is prime by Pocklington's theorem.