Prove or disprove $p_1p_2\cdots p_n+1$ is prime for all $ n\geq 1$ Let $p_1=2$, $p_2=3$, $p_3=5$ and, in general, let $p_i$ be the $i$-th prime. Prove or disprove that
$$p_1p_2 \cdots p_n+1$$
is prime for all $ n\geq 1$
Well I was able to find a counter example when $n=6$ but I do not have a general way to show why it shouldn't be prime.
 A: Maybe the best we can do is give an intuition based on the prime number theorem.
If $p_n$ is the $n$th prime, then the prime counting function $\pi(p_n) = n$.
Denote by $p_n\#$ the product of the first $n$ primes. Then, as you already know, $p_n\# + 1$ is not divisible by any of the first $n$ primes. This suggests that it is prime, contradicting, as you already know, the idea that the primes are finite. But if $p_n\# + 1$ is composite, as you already know, it also contradicts the idea that the primes are finite.
If $p_n\# + 1$ is indeed composite, its least prime factor must be greater than $p_n$ but less than $\sqrt{p_n\# + 1}$. Since there are something like $$\frac{\sqrt{p_n\# + 1}}{\log \sqrt{p_n\# + 1}}$$ primes less than $\sqrt{p_n\# + 1}$, there are about $$\frac{\sqrt{p_n\# + 1}}{\log \sqrt{p_n\# + 1}} - n$$ potential least prime factors for $p_n\# + 1$. And since this number is positive and greater than $1$ for $n > 3$, it seems likelier than not that $p_n\# + 1$ does indeed have a nontrivial least prime factor.
A: "For all" is not the same as "for almost all." The statement you were asked to prove or disprove says "for all $n \geq 1$," not "for all $n \geq 1$ with maybe one or two exceptions."
Then by finding a single $n$ that is a counterexample you have disproven the statement: $$2 \times 3 \times 5 \times 7 \times 11 \times 13 + 1 = 30031 = 59 \times 509.$$ That's it, that's not prime, you've done it.
Keep going a little bit further, though. You'll find that 9699691 is divisible by 347, 223092871 is divisible by 317, 6469693231 is divisible by 331, etc. In fact, take a look at Sloane's A051342: the counterexamples seem to outnumber the examples.
A: there's no general way, the $p_1p_2p_3...p_n+1$ expression is usually used to show that there are infinitely many prime numbers.
the number you get by that can be or not be a prime one depending on the situation.
