There are infinitely many prime numbers. Proof. We set a contrary of To prove this statement suppose,to the contrary,that there are our theorem and use finitely many primes.Then we may write these finitely many this assumption to primes in ascending order as start 2,3,5,,N, where N is the largest prime.Now consider the number M defined by Then we get a M=(2·3.5···N+1. contradiction through If M is prime,then M is a prime that is larger than the largest prime a valid argument N.Therefore,we must conclude that M is not prime,and so it is divisible by some prime number,P.However,P must appear inthe list of primes 2,3,5, which we gave earlier.But when we divide M by P,we obtain a Then we get the remainder of 1 Therefore,P cannot be a factor of M and we have theorem quickly contradicted our assumption that thre are finitely many primes. Thus,there cxist infinitely many primes. ■There are infinitely many prime numbers. We set a contrary of our theorem and use this assumption to start Then we get a contradiction through a valid argument Then we get the theorem quickly