If you read this carefully... One way to prove that there are infinitely many prime numbers is by using the concept of contradiction. Suppose that there are

If you read this carefully... One way to prove that there are infinitely many prime numbers is by using the concept of contradiction. Suppose that there are only a finite number of prime numbers, say p_1, p_2, ..., p_n. Then consider the number N defined as follows: N = p_1 * p_2 * ... * p_n + 1 If N is not prime, then it must have a prime divisor p that is not one of the primes p_1, p_2, ..., p_n. However, this contradicts the assumption that there are only finitely many prime numbers, since p is a prime that is not on the list. Therefore, N must be a prime number, which means that there are at least one more prime number beyond the list p_1, p_2, ..., p_n. This contradiction shows that our assumption that there are only finitely many prime numbers must be false, so there must be infinitely many prime numbers. It's right! But there's a pointless appeal to the primality or non-primality of N. That can be cut out.