Prove that there exists integer n such that n n 1 n 2 n

Prove that there exists integer n such that n, n + 1, n + 2,.. ., n + 14, n + 15 are all composite.

Solution

Let n =18! (factorial 18) +2

Consider n,n+1,............n+15

n is divisible by 2

n+1 is divisible by 3

n+2 is divisible. by 4

.......

n+15 is divisible by 17.

so n, n+1,n+2,...........n+15 are consecutive composite numbers.