Goldbach's Conjecture

Goldbach's conjecture is a longstanding unsolved problem in number theory, posited by Christian Goldbach in a letter to Leonhard Euler in 1742. It asserts that every even natural number greater than 2 can be expressed as the sum of two prime numbers. Despite extensive computational verification up to 4×10^18 and various partial results supporting the conjecture, a formal proof remains elusive. The conjecture has two forms: the strong version, which pertains to even integers, and the weak version, which states that every odd integer greater than 5 can be expressed as the sum of three primes. Historical efforts to prove the conjecture have included significant contributions from mathematicians such as Hardy, Littlewood, and Chen Jingrun, among others. The conjecture has also inspired cultural references, appearing in literature and film. Modern interpretations of Goldbach's conjecture continue to explore its implications and relationships with other mathematical theories, including the generalized Riemann hypothesis. While the weak conjecture has been proven, the strong conjecture remains an open question in mathematics.

- Goldbach's conjecture states that every even integer greater than 2 is the sum of two primes.

- The conjecture has been verified for even numbers up to 4×10^18 but lacks a formal proof.

- The weak version of the conjecture, stating that every odd integer greater than 5 can be expressed as the sum of three primes, has been proven.

- Historical contributions to the conjecture have come from notable mathematicians, including Euler and Hardy.

- Goldbach's conjecture has influenced popular culture, appearing in various literary and cinematic works.