Goldbach's Conjecture says that all even integers greater than can be written as a sum of prime numbers (may or maynot be distinct).
- & are not equal to
We know that is always divisible by . It can easily be proven -
For all not equal to ,
Let , so let for some integer .
We know and
Taking on both sides,
would be .
Now () can be any number less than and we can assume that can also be any number less than keeping in mind that there are infinite primes and there is no sequence in the gap between any prime.
To keep in mind- Also, cannot be equal to because then, will have a prime factor.
One more fact is that for being a multiple of , it cannot be proven.
Hence, it is proven for and not equal to .
- or/and equal to
where one or both maybe 2. It is obvious that for being and any other prime, it will be an odd integer.
So, the only possibility is that and are equal to .
So, (See, can be written as sum of 2 primes as )
Hence, proved that both the primes have to be simultaneously, or both have to be .
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