TITLE: If $n = 51! +1$, then find number of primes among $n+1,n+2,\ldots, n+50$
QUESTION [13 upvotes]: If $n = 51! +1$, Then find no of primes among $n+1,n+2,\ldots, n+50$
Really speaking, I don't have any clue ...

REPLY [4 votes]: In addition to Michael Hardy's answer, which precludes all of n+1 to n+50, you can also prove that n is in fact composite as well using Wilson's Theorem which states that a number n is prime if and only if (n - 1)! ≡ (n-1) mod n. 
Here, n is (51! + 1), so (n-1)! = 51!!. Therefore, (51! + 1) is prime iff 51!! ≡ 51! mod (51! + 1). However, we now that (51!+1) is of course a factor of 51!! since (51! + 1) < 51!!, so it is impossible that 51!! would be congruent to anything other than 0 modulo (51! + 1). Therefore, 51! + 1 is also definitely not prime.