The Goldbach Conjecture states that "Every even integer greater than 2 can be written as the sum of two primes".
Fundamental theorem of Arithmetic states that "Every integer can be expressed in terms of powers of prime factors"
So, in order to prove the Goldbach Conjecture, couldn't we have used the fundamental theorem of arithmetic (Taking multiplication as repeated addition and power as repeated multiplication)?
For example, the number 24=11+13.
Taking 24=(2^3)*(3)=[(2+2)+(2+2)+(2+2)]+[(2+2)+(2+2)+(2+2)]+[(2+2)+(2+2)+(2+2)] Now if we could take one prime number out (since as per the fundamental theorem, we could take one prime number) and there is at least a probability of 1 to get another prime number to form a pair of prime numbers. [5,19],[11,13].
(It seems every even number N if split by a prime number A, the other part B need not be a prime number but there exists a prime number C which could add up A+C=N)