# Math Help - Prime Number Proof

1. ## Prime Number Proof

The problem states: If p1, p2, p3, ... , pn are the first n primes, show that p1*p2*p3...*pn + 1 is prime.

I'm guessing the proof has something to do with the fundamental theorem of arithmetic, that any number can be expressed as a product of primes. But how would I go about proving this?

2. Originally Posted by aaronrj
The problem states: If p1, p2, p3, ... , pn are the first n primes, show that p1*p2*p3...*pn + 1 is prime.

I'm guessing the proof has something to do with the fundamental theorem of arithmetic, that any number can be expressed as a product of primes. But how would I go about proving this?
This is false.
It happens to have a different prime divisor but it is not necessarily prime.

3. The smallest such example is $2\times3\times5\times7\times11\times13 + 1 = 30031 = 59\times509$.

4. ## Sieve of Eratosthenes

Now I understand why it is false. All of the primes up to the square root of 30031 are possible divisors. Are there any other quicker methods than this to find out if a number is prime?

5. Hello,
Originally Posted by aaronrj
Now I understand why it is false. All of the primes up to the square root of 30031 are possible divisors. Are there any other quicker methods than this to find out if a number is prime?
They are possible divisors, but they're not necessarily divisors !
That means that your property is not always false, nor always true.

Note that what you've said is similar to the proof that there are infinitely many primes.