Show that:
For the hypothesis it'd be enough to say that is a positive integer such that is not a prime number.
Let be the maximum power of a prime dividing .
- If is divisible by at least 2 primes, then and so appears as a factor of , thus
- Otherwise . If then appears as a factor of - this can be seen as in the previous part- so and so . Now if we have that is a factor of , but, since n is not prime, it must be that and so appears also as a factor in the product, because (*). Clearly so we have that
(*) iff and this holds since we were considering
From this analysis it follows that . (this holds for all the maximum powers of primes dividing n+1)
And therefore
Remark: This doesn't hold when is a prime greater than 2. ( it holds for )