The Goldbach Conjecture is one of the most famous unsolved problems in mathematics. Proposed by Christian Goldbach in 1742, it states that every even integer greater than 2 can be expressed as the sum of two prime numbers. For example, 4 equals 2 plus 2, 6 equals 3 plus 3, 8 equals 3 plus 5, and 10 can be written as either 3 plus 7 or 5 plus 5.
The Goldbach Conjecture has a rich historical background. It originated from correspondence between Christian Goldbach, a German mathematician born in 1690, and the famous Swiss mathematician Leonhard Euler, born in 1707. In 1742, Goldbach wrote a letter to Euler proposing what would become one of mathematics' most enduring unsolved problems.
Despite being proposed nearly 300 years ago, the Goldbach Conjecture remains unproven. However, extensive computational verification has been performed using modern computers. The conjecture has been verified for all even integers up to 4 times 10 to the 18th power, showing a 100% success rate. This massive computational effort using distributed computing gives strong evidence for the conjecture's truth, but a mathematical proof remains elusive.
The Goldbach Conjecture is part of a family of related problems in number theory. The Weak Goldbach Conjecture states that every odd number greater than 5 can be expressed as the sum of three primes. This was actually proven in 2013 by mathematician Harald Helfgott. Another famous related problem is the Twin Prime Conjecture, which proposes that there are infinitely many pairs of primes that differ by 2, such as 3 and 5, or 11 and 13. This conjecture remains unproven like the original Goldbach Conjecture.
The Goldbach Conjecture remains unsolved because it touches on fundamental mysteries of prime numbers. As numbers get larger, primes become increasingly sparse and their distribution appears irregular, making it difficult to guarantee that any even number can always be expressed as the sum of two primes. The conjecture's significance extends beyond the problem itself - attempts to solve it have advanced our understanding of sieve theory, analytic number theory, and computational mathematics. Despite nearly three centuries of effort by brilliant mathematicians, the Goldbach Conjecture remains one of mathematics' greatest unsolved mysteries.