Find the number of pairs of positive integers (n, k) that satisfy the equation $(n+1)^{k} -1 = n!$
My try: seeing the problem, I saw Wilson's theorem which states that $(p-1)! = p-1 mod p$. Therefore, we can let $a-1=n$, then $a^k -1 =(a-1)!$ where in A is prime, I tried a= 2, a=3, and a=5 as solutions but can't find anymore, why cant i find anymore primes?
