I want to if there is a proof for Goldbach Conjecture or any progress to prove it?

Answer 1

There is no proof yet. The following site lists the progress in its paragraph "Rigorous results":
http://en.wikipedia.org/wiki/Goldbach's_conjecture

Some results that have been proven:
- a sufficiently large even number can be written as the sum of either two primes, or a prime and the product of two primes
- "most" even numbers were expressible as the sum of two primes (meaning that below N there are only O(N^a) exceptions, a<1

Answer 2

In mathematics, Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states:

Every even integer greater than 2 can be written as the sum of two primes.
For example,

4 = 2 + 2
6 = 3 + 3
8 = 3 + 5
10 = 3 + 7 = 5 + 5
12 = 5 + 7
14 = 3 + 11 = 7 + 7
etc.
On 7 June 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII) in which he proposed the following conjecture:

Every integer greater than 2 can be written as the sum of three primes.
He considered 1 to be a prime number, a convention subsequently abandoned. So today, Goldbach's original conjecture would be written:

Every integer greater than 5 can be written as the sum of three primes.
Euler, becoming interested in the problem, answered with an equivalent version of the conjecture:

Every even integer greater than 2 can be written as the sum of two primes,
adding that he regarded this a fully certain theorem ("ein ganz gewisses Theorema"), in spite of his being unable to prove it.

The former conjecture is today known as the "ternary" Goldbach conjecture, the latter as the "strong" or "binary" Goldbach conjecture. The conjecture that all odd numbers greater than 9 are the sum of three odd primes is called the "weak" Goldbach conjecture. Both questions have remained unsolved ever since, although the weak form of the conjecture is much closer to resolution than the strong one.

As with many famous conjectures in mathematics, there are a number of purported proofs of the Goldbach conjecture, none of which are currently accepted by the mathematical community.

Because it is easily understood by laymen, Goldbach's conjecture is a popular target for pseudomathematicians who attempt to prove it, sometimes even disprove it, using only high-school-level mathematics. It shares this fate with the four-color theorem and Fermat's last theorem, each of which also has an easily stated problem, but a current proof which is extraordinarily elaborate.

It is possible that Goldbach's conjecture can yield to simple methods, but given the amount of professional attention paid to the conjecture, it is unlikely that a proof or a counter-example will be easy to find.

see this:
http://www.ieeta.pt/~tos/goldbach/help.html
http://www.utm.edu/research/primes/glossary/GoldbachConjecture.html
http://www.math.dartmouth.edu/~euler/correspondence/letters/OO0765.pdf

Answer 3

To add a bit to Cordefr's answer:
Goldbach's conjecture is known to be true for
all even numbers up to 10^20.
The weak Goldbach conjecture is known to
be true for all odd numbers greater than 10^43000.
There is an article in Mathematics of Computation
(sorry, can't remember the exact reference) that
gives lots of info on these conjectures.