In 1742, German mathematician Christian Goldbach wrote, in a letter to famed mathematician Leonhard Euler, that he believed “Every integer greater than two can be written as the sum of three primes.” In subsequent correspondence, the stronger version “Every even integer can be expressed as the sum of two primes” was suggested, as well as some other variants. The “odd” variant of the Goldbach conjecture is that every odd number greater than 7 can be expressed as the sum of three odd primes.
To this date, although extensive computer tests have found no counter-examples to these conjectures, no proofs are known. Deshoulliers, Effinger, te Riele and Zinoviev published a proof of the odd Goldbach conjecture, but it assumed the generalized Riemann hypothesis (another pre-emiment unsolved conjecture of mathematics), and so it is not really a proof of the original conjecture. Collectively these conjectures are among the oldest and most prominent unsolved problems in mathematics.
A few weeks ago (1 Feb 2012), the well-known Australian Fields medalist mathematician Terence Tao posted an arXiv paper entitled “Every odd number greater than 1 is the sum of at most five primes.” The title tells it all. Although numerous weaker results have been published in this area, Tao’s result, if it survives peer review by highly qualified mathematicians, would clearly be the strongest and most satisfactory yet.
A brief summary of prior results (taken from MathWorld) runs as follows:
The conjecture that all odd numbers
are the sum of three odd primes is called the “weak” Goldbach conjecture. Vinogradov (1937ab, 1954) proved that every sufficiently large odd number is the sum of three primes. (Nagell 1951, p. 66; Guy 1994), and Estermann (1938) proved that almost all even numbers are the sums of two primes. Vinogradov’s original “sufficiently large”
was subsequently reduced to
by Chen and Wang (1989). Chen (1973, 1978) also showed that all sufficiently large even numbers are the sum of a prime and the product of at most two primes (Guy 1994, Courant and Robbins 1996).
Tao presents a short synopsis of his wonderful result on his blog. He mentions that his paper utilizes the Hardy-Littlewood circle method, one of the most frequently employed techniques of analytic number theory, due to famed British mathematicians G. H. Hardy and J. E. Littlewood in the 1920s, but broadly based on Hardy’s earlier work with Indian mathematician Srinivasa Ramanujan.
While Terry Tao’s manuscript includes much impressive analysis of his own, he carefully notes that he relies on the results of numerous other contemporary mathematicians (the bibliography includes 39 references). Among the key results he utilizes are those of Jean Bourgain, Jing Run Chen, Xavier Gourdon, Ming-Chit Liu, Hugh Montgomery, Harmut Siebert, Ivan Vinogradov and Tianze Wang. As Newton, in a moment of unusual candor, once confessed, “If I have seen further it is by standing on ye sholders of Giants.”