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

Question

Recently I came across this proof for fact that primes are infinite.

It's a proof by contradiction. The proof assumes that primes are finite and there is a prime M which is larger than any prime out there. Then you basically take the product of all primes up to and including M and you add one to it, which the source where I read this proof claims is a prime obviously larger than M hence a contradiction to our assumption that primes are finite.

So why is it that product of first N primes + 1 another prime?

It doesn't have to be prime (and usually isn't). It is divisible by some prime (because all natural numbers $>1$ are and that prime has to be bigger than anything on your list. — lulu, Jul 31 '15 at 19:18
consider $P=p_1 \cdot p_2\cdot p_3...p_n+1$ now do case work — Dr. Sonnhard Graubner, Jul 31 '15 at 19:18
The proof doesn't say that the product plus $1$ is prime! It says that any prime factor of the product plus $1$ cannot be one of the first $N$ primes... — David C. Ullrich, Jul 31 '15 at 19:18
It is not the case that the product of the first $N$ primes + 1 is another prime, only that it contains a novel prime factor. — hardmath, Jul 31 '15 at 19:19
Not necessarily another primes, but most definitely divisible by another prime. — barak manos, Jul 31 '15 at 19:20
My bad. I read this proof long back and back then I assumed this is what the author meant! — user2340452, Jul 31 '15 at 19:22
To be concrete, $2⋅3⋅5⋅7⋅11⋅13+1 = 59\cdot 509$ is not a prime. — hardmath, Jul 31 '15 at 19:26

score 22 · Answer 1 · answered Jul 31 '15 at 19:25

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).

score 7 · Answer 2 · answered Jul 31 '15 at 19:24

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?

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