Suppose there are only $n$ primes, $p_1,...,p_n$ and let $M=p_1...p_n+1$. If $M$ is composite, then there exists some prime $p_i$ which divides $M$. That is, there exists some $k$ such that $M=p_ik$. Note also that $p_i$ divides $M-1$. Therefore, there is some $\ell$ such that $M-1=p_i\ell$. Hence, we can write $M=p_i\ell+1=p_ik$. Alternately, $p_i(k-\ell)=1$. But this implies that $p_i$ divides $1$ and this is impossible. Therefore, the original assumption, that there are only $n$ primes, $p_1,...,p_n$, must be false.

Why is it that the product of first N prime numbers + 1 another prime?
I have multiple mobile devices (2x iPhone, 2x iPad) in my home, all which share the same iTunes account between them.  When iMessage is used to exchange messages between two of the devices (e.g. the two iPhones), the messages appear to be sent from and received by the same identity.  The only way I have found to correct this by turning off iMessage on one of the devices and only using SMS.

Is there a way to associate iMessage with a different iTunes account than other apps on the same device?
