View source for Goldbach problem

One of the well-known problems in number theory: To give a proof that every odd integer equal to or larger than 7 can be written as the sum of three prime numbers. It was posed in 1742 by Ch. Goldbach in a communication to L. Euler. Euler replied by pointing out that in order to solve this problem it is sufficient to prove that every even number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044550/g0445501.png" /> is the sum of two prime numbers. All attempts to solve the problem remained unsuccessful for a long time. G.H. Hardy and J.E. Littlewood in 1923 succeeded in showing that if certain theorems concerning [[Dirichlet-L-function|Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044550/g0445502.png" />-function]] (which have not been proved till now) are valid, then any sufficiently large odd number is the sum of three prime numbers. I.M. Vinogradov in 1937 discovered a new method in analytic number theory — the method of estimating trigonometric sums involving prime numbers — and applied this method to prove an asymptotic formula for the number of representations of odd numbers as sums of three prime numbers (cf. [[Vinogradov method|Vinogradov method]]; [[Trigonometric sums, method of|Trigonometric sums, method of]]). This formula implies that each sufficiently large odd number is the sum of three prime numbers. This is one of the major achievements of modern mathematics. Vinogradov's method offers a way for solving several problems of a much more general nature. The problem of writing every even number as a sum of two prime numbers has not yet (1989) been solved.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,   "The method of trigonometric sums in the theory of numbers" , Interscience  (1954)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Karatsuba,   "Fundamentals of analytic number theory" , Moscow  (1975)  (In Russian)</TD></TR></table>



====Comments====
In [[#References|[a1]]] a concise proof can be found of Vinogradov's result on the ternary Goldbach problem. In 1973 Chen Jing-run proved that every sufficiently large even number is a sum of a prime and a number composed of at most two primes. For a proof see [[#References|[a2]]], Chapt. 11.

In [[#References|[a3]]] Goldbach's conjecture (in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044550/g0445503.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044550/g0445504.png" />) is studied from a numerical point of view. Special attention is devoted to the smallest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044550/g0445505.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044550/g0445506.png" />. It was found that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044550/g0445507.png" /> is the worst case in the range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044550/g0445508.png" />, i.e. this number can be written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044550/g0445509.png" /> whereas every other <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044550/g04455010.png" /> not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044550/g04455011.png" /> has a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044550/g04455012.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044550/g04455013.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.C. Vaughan,   "The Hardy–Littlewood method" , Cambridge Univ. Press  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Halberstam,   H.-E. Richert,   "Sieve methods" , Acad. Press  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Granville,   J. van der Lune,   H.J.J. te Riele,   "Checking the Goldbach conjecture on a vector computer"  R.A. Molin (ed.) , ''Number theory and applications (Proc. First Conf. Canadian Number Theory Assoc., Banff, April 1988)'' , Kluwer  (1989)  ((Also: Mathematical Centre Report NM R8812 (1988)))</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  Wang Yuan (ed.) , ''Goldbach conjecture'' , World Sci.  (1984)</TD></TR></table>