How do you prove that if the sum of two prime numbers is prime then one of the numbers is 2?

Answer 1

We know that a Prime number is a positive integer greater than 1, whose divisors are 1 and itself. We know that the only even prime number is 2. That means that all other prime numbers are odd numbers.

We know that when we add two odd numbers the result is an even number, which are not prime numbers (expect 2, and 2 = 1 + 1 where 1 is odd but is neither prime nor composite). Thus adding two odd prime numbers cannot give us another prime number. We show that the conclusion follows from the premise:

Assume that r = p + q where all r, p and q are prime numbers, then we have that ris either even or not:

* If r is even then r is at least 4 (the smallest number which is the sum of two primes) and thus not a prime number. This contradicts the assumption that r is a prime number, and therefore we conclude that r is not even. * If ris odd then either p or q must be odd and the other one must be even, since both p and q are prime numbers one of them must be 2 (the only even prime number). 2 + 3 = 5, 2 + 17 = 19, are examples of such numbers. See http://en.wikipedia.org/wiki/Twin_prime for more.