Why is it that the product of first N prime numbers + 1 another prime? [duplicate]

Solution 1:

It is not necessarily another prime, but it is most definitely divisible by another prime.

For example, take $2\cdot3\cdot5\cdot7\cdot11\cdot13+1=30031$, which is not prime but divisible by $59$.

It cannot be divisible by any one of those first $N$ primes, so it has to be divisible by some other prime (or possibly, a prime by itself, which also qualifies under the same definition).

Solution 2:

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.