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There is no largest prime. In fact the classic proof that there are infinitely many primes starts with the assumption that there is a largest prime and proceeds
by jhart3333 13y ago
There is no largest prime. In fact the classic proof that there are infinitely many primes starts with the assumption that there is a largest prime and proceeds to use this number to construct a larger number which is also prime. This is a contradiction: "this is the largest prime" and "this is not the largest prime." So the original assumption must be false and "there is no largest prime" or equivalently "there are infinitely many primes". Recently it has been shown that this proof is a result of misunderstanding the one given in Euclid's Elements. Nevertheless it's still a fine proof.
- Zenst 13y agoThe largest known prime I was refering too was what is known now - http://en.wikipedia.org/wiki/Largest_known_prime_number http://en.wikipedia.org/wiki/Largest_known_prime_number So given that to know with all certaintity that only upto that value three times would be the only provable maximum value, until a new know prime is found and somebody invents a formular to drop in N and get the Nth prime out without waiting more than few seconds. Currently we can not do that at all, let alone that dream of many, maybe one day in our lifetimes. I agree, it is a fine proof. I hope to appreciete it more over the following days. Lots to go over.
- jhart3333 13y agoIt's been a while since I studied this stuff and I screwed up a crucial part of the argument. The larger number isn't guaranteed to be prime. It's either prime or composite. In either case you end up with a prime not in the finite list of primes that follow from the assumption of a largest prime. Here is a link to a good explanation that includes the composite case: http://www.jimloy.com/number/primes.htm http://www.jimloy.com/number/primes.htm