Thread: prime numbers

In proving that there are infinitely many primes, one could define a function by and show that (1) for all , (2) is either prime or composite and (3) has prime factors greater than if is composite.

So in general can we find a function defined by and choose accordingly to satisfy the above conditions? Could we define a function in another way that satisfies the above condtions?

keep in mind that n! +a where a is between 2 and n
are always composite numbers

Uggg, just never mind. No helpful thoughts... I'm thinking some more.