# prime numbers

• Sep 11th 2008, 09:32 PM
particlejohn
prime numbers
In proving that there are infinitely many primes, one could define a function $\displaystyle f: \mathbb{N} \to \mathbb{N}$ by $\displaystyle f(n) = n!+1$ and show that (1) $\displaystyle f(n) > n$ for all $\displaystyle n \in \mathbb{N}$, (2) $\displaystyle f(n)$ is either prime or composite and (3) $\displaystyle f(n)$ has prime factors greater than $\displaystyle n$ if $\displaystyle f(n)$ is composite.

So in general can we find a function $\displaystyle f: \mathbb{N} \to \mathbb{N}$ defined by $\displaystyle f(n) = n! + a$ and choose $\displaystyle a$ accordingly to satisfy the above conditions? Could we define a function $\displaystyle f: \mathbb{N} \to \mathbb{N}$ in another way that satisfies the above condtions?
• Sep 14th 2008, 02:17 PM
jbpellerin
keep in mind that n! +a where a is between 2 and n
are always composite numbers
• Sep 15th 2008, 09:47 PM
Jameson
Uggg, just never mind. No helpful thoughts... I'm thinking some more.