There is no largest prime. There are an infinite number of primes. Remember, an integer is either itself prime or can be uniquely factored into a product of primes. So assume that there were a largest prime P. Now form the product of all primes upto and including prime P, write 2*3*5*7*11*13.....*P = N. Add one, giving M=N+1. Now M is either prime or factorable. If it is prime, it is bigger than P, thus leading to a contradiction. Assume it is factorable, then if you divide M by any of the primes 2 through P there will always be a remainder (1) left over. So M must have a factor larger than P, which is a contradiction again. Therefore there is no largest prime P
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