Can anyone tell me the strengths and weaknesses of Goldbach's Conjecture?

Question

On 7 June, 1742, Christian Goldbach wrote a letter to Leonhard Euler, stating a conjecture he discovered. In this conjecture, he stated that every integer greater than 2 can be written as the sum of three primes. Note that at that time, Goldbach considered 1 to be a prime, a convention no longer followed, so the present equivalent of his conjecture would be: every integer greater than 5 can be written as the sum of three primes. This conjecture is known today as the “ternary” Goldbach’s conjecture.
Upon receiving this conjecture, Euler became interested in the problem and replied with a similar conjecture: every integer greater than 2 can be written as the sum of two primes. This conjecture is known today as the “strong” or “binary” Goldbach’s conjecture.

Answer 1

I think you are asking this question in the context of the Goldbach conjecture being called the "strong" Goldbach conjecture.

In fact, there is a "weak" Goldbach conjecture and a "strong" Goldbach conjecture. The weak conjecture says that every odd number greater than 7 can be written as the sum of 3 odd primes. The strong conjecture is the one you cited, that every integer greater than 2 can be written as the sum of 2 primes.

The "strength" of the strong conjecture is that it claims that more is true than the weak conjecture. In fact, if the strong conjecture is proven to be true, the the weak conjecture can automatically be proven to be true. In this sense, it is a bolder conjecture.

It is common in mathematics to have a weak and a strong form of a theorem or conjecture, with the strong form containing more information, and usually turning out to be harder to prove.

Answer 2

A well known weakness is that, unlike Riemann's hypothesis, it has never been linked to anything significant in other areas of maths.

Answer 3

What do you mean by strengths? This conjecture for one has been experimentally verified up to some considerably huge integer. Please elaborate more on your question.