I am trying to understand Euclid's proof that there is no largest prime number.
Suppose I assume that 97 is the largest prime number. I read that the proof involves multiplying all known primes together and then adding 1.
However, I am confused. If I take a number and add 1, the result is not always prime. For example, I can get a number like 64, which is not prime.
My question is: In Euclid's proof, why does multiplying all the assumed primes together and adding 1 show that there must be another prime number? If the resulting number is not prime, how does the proof still work?
I would appreciate a simple explanation.