One thing puzzled me is that, despite its simple form, I have not seen any intuitive reasons for Goldbach conjecture to be true.
Typical heuristic reason is based on probability arguments. Such arguments basically say that, since there are so many primes out there, so very likely, you will find a Goldbach combination for any even number (2n = p1 + p2).
Circle methods and sieve methods essentially follow above argument.
The issue with this argument is that, it can not explain why Goldbach conjecture hold true for small numbers. For example, why is it true for all even numbers less than 100 or 1000.
Are there any intuitive reasons for Goldbach conjecture to be true ?