There is a simple counterexample: the number 1:
- 1 is an odd number
- the first prime is 2 (not 1, see below) which is bigger than 1 so 1 cannot possibly be the sum of two primes.
There are plenty of other counterexamples:
- The sum of two odd numbers is even;
- All prime numbers except 2 are odd;
- When adding two prime numbers together, to get an odd result one of them must be even, namely 2;
- So any odd number that is 2 more than a composite number will not be expressible as the sum of two primes.
- examples: 11, 17, 23, 27, 29, 35, 37, ...
Another counterexample is the number 3:
- 3 is an odd number
- 3 can only be made by 2 + 1
- 1 is not a prime (see below)
A Prime number is a number that has exactly 2 distinct (different) factors.
The number 1 has only 1 distinct factor (the number 1) and so is not a prime number; the first prime number is 2.
