The Sum of Two Prime Numbers Greater than 2 is Even

In the previous posts, we have gone quite deep in delving and proving some complicated theorems. In this post, we go again the the basics. In this short proof, we show that the sum of two prime numbers, both greater than 2 is even.

Theorem

The sum of two primes, both greater than 2, is always even.

Proof

Let p and q be prime numbers both greater than 2. Then, both of them are odd numbers. This means that we can let p = 2r + 1 and q = 2s + 1 where r and s are positive integers.  Adding, we have

p + q = (2r + 1) + (2s + 1) = 2r + 2s + 2

Factoring out 2, we have

2s + 2r + 2 = 2(s + r + 1)

Since 2(s + r + 1) is divisible by 2, it is even. Therefore,

p + q = 2(s + r + 1)

is even.

 

Therefore, the sum of two primes both greater than 2 is even.

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