Why does Euclid theorem fail in some cases?

Question

The euclidean theorem says that if we have a limited prime numbers and we added 1 it cant be divided by any prime numbers

I notice that it work in some cases with lower number but when I added a couple of them things turn off.

M=2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 =9.699.690 => Then we plus +1 => M= 9.699.691

When I use this website to corroborate if it is a prime number it says that it is not.

I know there are infnite primes numbers. But why does this theorem have another conditions. Or it does sometimes doesn't work?

Thank you

Answer 1

That step of the proof is not saying that $M$ is prime. It's only saying that $M$ isn't divisible by any of the prime numbers in your finite list. Since all numbers are divisible by some prime numbers that allows you to add a prime to the list, whether or not $M$ is that prime.

In the case of the list of primes you used there, the argument goes:

Is $\{2,3,5,7,11,13,17,19 \}$ the list of all primes?

No. $2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17 \times 19 + 1 = 9699691$ is a number that is not divisible by any of our list of primes. $9699691$ is divisible by the prime $347$. $347$ is a prime number that isn't in our list.

Answer 2

“If we had limited prime numbers” then the product of all primes, plus 1, would have no prime factor and therefore would be prime.

If you multiply all primes from 2 to P and add 1, that number has indeed no prime factor up to P. But the prime numbers are not limited, so it can have factors greater than P.