Primality proof for n = 112600039977335252189018616651067202354555503111433:

Take b = 2.

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

718995365596683143 is prime.
b^((n-1)/718995365596683143)-1 mod n = 15865270877330744392360927023167561433727193543917, which is a unit, inverse 72854732329452665027269890060225838476444883262680.

124778574733515931 is prime.
b^((n-1)/124778574733515931)-1 mod n = 3538827383310846898460923226246993512485808845259, which is a unit, inverse 69888733089952294658870157338184018491612939480748.

(124778574733515931 * 718995365596683143) divides n-1.

(124778574733515931 * 718995365596683143)^2 > n.

n is prime by Pocklington's theorem.