Is there a prime number that is the largest? The answer is \"no.\" The proof was

Question

Is there a prime number that is the largest? The answer is "no." The proof was given by Euclid (Proposition 20, Book IX) in his Elements. Here we follow Euclid's argument with the modern notation. We write the primes 2,3,5,7,... in ascending order. For any prime p. we want to prove that there is a new prime q such that q > p. To do this, we write N = (2 . 3 . 5 . 7... p) + 1. If N is prime, we are done. Otherwise, N is divisible by some prime number q. Prove that such q is the desired prime number.

Explanation / Answer

Any prime less than or equal to p is a factor of the term (2 · 3 · 5 · 7 · ... · p) Thus by inspection of the equation, none of those primes can be a factor of N because they have a remainder of 1* Thus any existing prime factor of N must be greater than p * If this isn't clear, try thinking of it this way; (2 · 3 · 5 · 7 · ... · p) is an exact multiple of any given prime up to p. Thus, adding 1 will not be an exact multiple. And N is gotten by adding 1

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