# Prove that the sum of two odd primes will never result in a prime?

## Main Question or Discussion Point

How can I prove that the sum of two odd primes will never result in a prime?

Would this be proof?:

The sum of two odd primes will sometimes result in a prime.
This is true because 2 + 3 = 5, which is a prime.

So since this is true, does this proof the situation? Thank you.

Tom Mattson
Staff Emeritus
Gold Member

Originally posted by Caldus
How can I prove that the sum of two odd primes will never result in a prime?
Here's something to get you started:

Write down a general expression for two different odd numbers {x,y} in terms of integers {m,n}, respectively:

x=2m+1
y=2n+1

The sum is always even. Now you're not out of the woods yet, because 2 is even and it is also a prime. So, you have to use the above (along with the definition of "prime") to show that, if x+y=2, then at least one of the numbers must be nonprime (that is, 1 or an odd negative number).

Would this be proof?:

The sum of two odd primes will sometimes result in a prime.
This is true because 2 + 3 = 5, which is a prime.

So since this is true, does this proof the situation? Thank you.
No, that does not prove it. Proof by contradiction means that you assume the negative of the statement you are trying to prove and show an absurdity.

Also, you did not stick to the condition stipulated by the statement, which specifies odd primes (that means you can't use 2).

The sum of two odd primes will sometimes result in a prime.
This is true because 2 + 3 = 5, which is a prime.
That's not a proof by contradiction, that's a proof by counterexample (but as Tom said, the counterexample isn't valid).

Tom Mattson
Staff Emeritus