Primality proof for n = 1359580455984873519493666411:

Take b = 2.

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

23538849507889 is prime.
b^((n-1)/23538849507889)-1 mod n = 1318117154905851161536155703, which is a unit, inverse 1103942428099668499562139506.

44774420161 is prime.
b^((n-1)/44774420161)-1 mod n = 690367073390889282742820561, which is a unit, inverse 318699155462093658311429229.

(44774420161 * 23538849507889) divides n-1.

(44774420161 * 23538849507889)^2 > n.

n is prime by Pocklington's theorem.