Primality proof for n = 2261713:

Take b = 2.

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

47119 is prime.
b^((n-1)/47119)-1 mod n = 1290808, which is a unit, inverse 2224341.

(47119) divides n-1.

(47119)^2 > n.

n is prime by Pocklington's theorem.