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Proof that there is unlimited prime numbers

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It's not that p1\*p2\*...\*pn+1 is necessarily prime, it's that it must be divisible by some prime q, and that q can't be any of the primes p1,p2,...,pn.
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You need to be a little tighter. You start by assuming that there is a finite number of primes, and that p1, p2, …pn is a list of _all_ the primes—that’s it, there are no more. Then you show that that assumption can’t be true, because there must be another prime that wasn’t in your list. And even if you add this new prime to your list, there must be yet another, and so on…
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If I remember correctly, it was case by case. If the q= p1*… *pn+1 is prime then uve found a larger prime than any of the primes.

The other case is that q is not a prime number.
If it’s not a prime number then there exists some number that q is divisible by some number c other than 1 and itself.
More importantly, c does not equal any of p1 to pn.
Because suppose c did equal a prime number from p1 to pn, then what happens when you divide q by c?
(p1 *…* pn +1) / c= (p1 *…* pn)/c  + 1/c = g+1/c.
Since c is one of p1to pn, you can tell that g will just be the product of all primes except whatever prime that c was equal to. Such g will be a natural number since it’s just product of natural numbers, but then you have 1/c and g +1/c will not be a natural number. So c as one of the primes already in the p1 to pn is not a valid choice as a divisor. Then the case 2 also reveals that c is any of p1to pn. Again in case 2 you find a prime that is not in the finite set of all primes.

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