The set of prime numbers includes only one even number, 2. The product of consecutive primes will always be even, so adding 1 will make the result odd. Since all the primes up to n are included in the product, then adding 1 means that none of them will divide into this final sum, because they will all have a remainder of 1 when dividing by any one of them. For example, the product of 1, 2, 3, 5, 7, 11 is 2310. Add 1 and we have 2311 which cannot be divisible by any of the constituent primes of 2310. If we take any number X, we know that it is only necessary to attempt to divide it by a prime up to √X to discover whether X itself is prime. If X=2311, √2311 is about 48. But the constituent primes of 2310 only go up to 11, and there are 10 more primes less than 48, and so are potential candidates as factors. It is possible, then, that the number produced by the product of n primes plus 1 is not prime because there is at least one prime number that has not been included in the product. So the statement may not be true. As it happens, 2311 is a prime number.
Now, take the products up to 13. This gives us 30030. Add 1 to get 30031, which is not exactly divisible by any primes up to 13. √30031=173 approx. There are many primes between 17 and 173, and, in fact, 30031=59*509. This proves that the statement is not true. As the product of primes increases it's clear that more and more prime factors become potential factors.
