Proof of Goldbach’s Conjecture Reza Javaherdashti farzinjavaherdashti@gmail.com Abstract After certain subsets of Natural numbers called “Range” and “Row” are defined, we assume (1) there is a function that can produce prime numbers and (2) each even number greater than 2, like A, can be represented as the sum of n prime numbers. We show this by DC(A) ≤ n. Each Row is similar to each other in properties,(so is each Range). It is proven that in an arbitrary Row for any even number greater than 2, DC(A)=2, that is to say, each even number greater than two is the sum of two prime numbers. So Goldbach’s conjecture is proved. 1.Historical Background: Of still-unsolved problems on prime numbers one can mention Goldbach’s conjecture. Goldbach (1690-1764) in his letter to Euler in 1742, asked if any even number greater than 2 could be written as the sum of two prime numbers. Euler could not answer nor could he find any counter-example. The main problem with Goldbach’s conjecture is that in most of theorems in arithmetic, prime numbers appear as products, however, in Goldbach’s conjecture it is the addition of prime numbers that makes all the problem. In 1931, a young, not that famous Russian mathematician named Schnirelmann (1905-1938) proved that any positive integer could have been represented, at most, as the sum of 300,000 prime numbers. The reasoning was constructive and direct without giving any practical use to decompose a given integer into the sum of prime numbers. Some years after him, the Russian mathematician Vinogradoff by using and improving methods invented by the English mathematicians, Hardy and Littlewood, and their great Indian colleague Ramanujan, could decrease number of the mentioned prime numbers from 300,000 to 4. Vinogradoff’s approach has been proved to be true for integers “large enough”. With exact words, Vinogradoff proves that there exists an integer like N so that for any integer n>N, it can be represented, at most, as the sum of four prime numbers. He gives noway to determine and measure N. Vingoradoff’s method has actually proved 1