The conjecture is actually worded a bit differently.

Every even number greater than 2 can be written as the sum of two prime numbers.

The initial wording of the conjecture included 2 as a number that could be written as a sum of two prime numbers but that was also assuming 1 was a prime number.

4 is the first applicable number of the conjecture. It is also the only number than can have it's sums expressed with even numbers. 2 is the only even prime number. If you subtract 2 from an even number you get an even number which means it is at least divisible by 2, thus not a prime number.

## The conjecture's wording

The conjecture is actually worded a bit differently.

Every even number greater than 2 can be written as the sum of two prime numbers.

The initial wording of the conjecture included 2 as a number that could be written as a sum of two prime numbers but that was also assuming 1 was a prime number.

4 is the first applicable number of the conjecture. It is also the only number than can have it's sums expressed with even numbers. 2 is the only even prime number. If you subtract 2 from an even number you get an even number which means it is at least divisible by 2, thus not a prime number.