# Prove that every number between two factors of primes is composite.

I am looking for some help with this problem:

Let p1, p2, ... , pn+1 be the first n+1 primes in order. Prove that every number between p1 * p2 * p3 ... pn + 1 and p1 * p2 * p3 * ... * pn+1 -1 is composite (inclusive of the second term). How does this show that there are gaps of arbitrary length in the sequence of primes?

I know that if p1 * p2 * p3 ... pn + 1 is not prime, it must have a prime factor larger than pn, and I am guessing this can be leveraged to prove the above problem, but I am not sure where to start or how to put this in mathematical terms. Any help/hints would be great.

EDIT: Sorry I transcribed the problem wrong. This is the correct problem:

Let p1, p2, ... , pn+1 be the first n+1 primes in order. Prove that every number between p1 * p2 * p3 ... pn + 1 and p1 * p2 * ... * pn + pn+1 -1 is composite (inclusive of the second term). How does this show that there are gaps of arbitrary length in the sequence of primes?

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Thank you for noticing this. I had transcribed the problem wrong. Please see the edits. –  Alex Oct 28 '13 at 2:51

The usual way that the final result is proved is by considering the interval $[n!+2, n!+n]$, the terms of which are divisible by $2,3,\ldots, n$, respectively and therefore composite.
Check divisibility by $p_1, p_2,\ldots, p_n$ and see! –  vadim123 Oct 28 '13 at 3:03