
The answer is, given the assumptions here, P must not be prime.

And the reason for that is P is greater than Pn.

And we said this set includes all prime numbers

that's the assumption we started with for this proof,

and this number's the product of many numbers

all of these are positive, adding 1 to it.

So the value of P must be greater that P sub N

and that means, according to our assumption about the limited set of all primes,

P must not be prime.

So since P is not a prime, that means it must be a composite

which means it must be the product of some prime number and some other integer.

So that means we can write P as some prime,

and we've said all the primes are in this set,

so it's something selected from that set,

multiplied by some other numberwe don't know what Q is

other than it must be an integer.

So now we have P, which we computed as the product of all these primes plus 1

which is equal to some prime from that set times Q.