Prove sum of two primes is even

[cragar]

Homework Statement

Prove: If a and b are prime numbers larger than 2, then a + b is even.

The Attempt at a Solution

Can i just say that prime numbers larger than 2 are odd and then prove that the sum of 2 odd numbers is even. And can i say that prime numbers larger than 2 are odd because prime numbers only have factors of 1 and themselves and if 2 was a factor then it wouldn't be prime.

 

[phinds]

Can i just say that prime numbers larger than 2 are odd and then prove that the sum of 2 odd numbers is even. And can i say that prime numbers larger than 2 are odd because prime numbers only have factors of 1 and themselves and if 2 was a factor then it wouldn't be prime.

That's what I would do, but then I'm an engineer, not a mathematician

 

[micromass]

Hi cragar!

What you say is certainly correct. But I don't know how rigourous your argument needs to be. For example, you still need to show that the sum of two odd numbers is even.

I think the trick is here to write it down in a clear, organized way.

 

[cragar]

ok so would I show that the sum of 2 odd integers is even by
proof:
Let x and y be odd integers and there exists integers a and b such that
x=2a+1 and y=2b+1 .
then x+y=(2a+1)+(2b+1)
then x+y=2a+2b+2
x+y=2(a+b+1)
since x+y is divisible by 2 therefore it is even.

 

[micromass]

Looks good!

 

[cragar]

sweet , thanks for your help.

 

[cragar]

That's what I would do, but then I'm an engineer, not a mathematician

Ha ha, you I am a physics major learning to write proofs. I can't tell you how many times my physics profs cut corners on the math that would make a mathematician cringe.

 

[osnarf]

You should try an engineering class, :P. It makes ME cringe.

 

[Mark44]

ok so would I show that the sum of 2 odd integers is even by
proof:
Let x and y be odd integers and there exists integers a and b such that
x=2a+1 and y=2b+1 .
then x+y=(2a+1)+(2b+1)
then x+y=2a+2b+2
x+y=2(a+b+1)
since x+y is divisible by 2 therefore it is even.

Nothing wrong with what you did, but you can economize a bit like so:
then x+y=(2a+1)+(2b+1) = 2a+2b+2 = 2(a+b+1)
Since x+y is divisible by 2, therefore it is even.