Goldbach Problem


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Goldbach Problem

 

one of the well-known problems in the theory of numbers. The problem consists in proving that any integer equal to or greater than 6 can be represented in the form of the sum of three prime numbers. This problem was posed by C. Goldbach in 1742 in a letter to L. Euler. In his reply, Euler pointed out that in order to solve the problem, it is sufficient to prove that any even number is the sum of two prime numbers. For a long period of time it was impossible to find any way of examining the Goldbach problem. In 1923, G. Hardy and D. Littlewood were able to show that if certain theorems are valid (and these have not yet been proved) for the so-called Dirichlet L-series, then any sufficiently large odd number is the sum of three prime numbers. A major advance in the solution of the Goldbach problem was the theorem proved by L. G. Shnirel’man (1930) that any integer greater than 1 is the sum of a limited number of prime numbers. In 1937. I. M. Vinogradov proved that any sufficiently large odd number is the sum of three prime numbers, that is, in essence solved the Goldbach problem for odd numbers. This has been one of the major achievements in contemporary mathematics. The method created by I. M. Vinogradov in solving the Goldbach problem has made it possible to solve a number of essentially more general problems. Another proof of the theorem concerning the representation of a sufficiently large odd number in the form of the sum of three prime numbers was given in 1945 by lu. V. Linnik. The problem of separating an even number into a sum of two prime numbers has still not been solved.

REFERENCES

Vinogradov, 1. M. “Metod trigonometricheskikh summ v teorii chisel.” Tr. Matematicheskogo in-ta AN SSSR, 1947, vol. 23.
Chudakov. N. G. “O probleme Gol’dbakha.” Uspekhi matemati-cheskikh nauk, 1938. issue 4.