Goldbach's weak conjecture
In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that This conjecture is called "weak" because if Goldbach's "strong" conjecture (concerning sums of two primes) is proven, then this would also be true. For if every even number greater than 4 is the sum of two odd primes, adding 3 to each even number greater than 4 will produce the odd numbers greater than 7 (and 7 itself is equal to 2+2+3). In 2013, Harald Helfgott  published a proof of Goldbach's weak conjecture. As of 2018, the proof is widely accepted in the mathematics community, but it has not yet been published in a peer-reviewed journal. The proof was accepted for publication in the Annals of Mathematics Studies series in 2015, and has been undergoing further review and revision since. Some state the conjecture as This version excludes 7 = 2+2+3 because this requires the even prime 2. On odd numbers larger than 7 it is slightly stronger as it also excludes sums like 17 = 2+2+ 13, which are allowed in the other formulation. Helfgott's proof covers both versions of the conjecture. Like the other formulation, this one also immediately follows from Goldbach's strong conjecture. Origins. The conjecture originated in correspondence between Christian Goldbach and Leonhard Euler. One formulation of the strong Goldbach conjecture, equivalent to the more common one in terms of sums of two primes, is The weak conjecture is simply this statement restricted to the case where the integer is odd (and possibly with the added requirement that the three primes in the sum be odd). Timeline of results. In 1923 , Hardy and Littlewood showed that, assuming the generalized Riemann hypothesis, the weak Goldbach conjecture is true for all sufficiently large odd numbers. In 1937, Ivan Matveevich Vinogradov eliminated the dependency on the generalised Riemann hypothesis and proved directly (see Vinogradov's theorem) that all sufficiently large odd numbers can be expressed as the sum of three primes. Vinogradov's original proof, as it used the ineffective Siegel–Walfisz theorem, did not give a bound for "sufficiently large"; his student K. Borozdkin (1956) derived that formula_1 is large enough. The integer part of this number has 4,008,660 decimal digits, so checking  every number under this figure would be completely infeasible. In 1997, Deshouillers, Effinger, te Riele and Zinoviev published a result showing that the generalized Riemann hypothesis implies Goldbach's weak conjecture for all numbers. This result combines a general statement valid for numbers greater than 10 with an extensive computer search of the small cases. Saouter also conducted a computer search covering the same cases at approximately the same time. Olivier Ramaré in 1995 showed that every even number "n" ≥ 4 is in fact the sum of at most six primes, from which it follows that every odd number  n ≥ 5 is the sum of at most seven primes. Leszek Kaniecki showed every odd integer is a sum of at most five primes, under the Riemann Hypothesis. In 2012, Terence Tao proved this without the Riemann Hypothesis; this improves both results. In 2002, Liu Ming-Chit (University of Hong Kong) and Wang Tian-Ze lowered Borozdkin's threshold to approximately formula_2. The exponent is still much too large to admit checking all smaller numbers by computer. (Computer searches have only reached as far as 10 for the strong Goldbach conjecture, and not much further than that for the  weak Goldbach conjecture.) In 2012 and 2013, Peruvian mathematician Harald Helfgott released a pair of papers improving major and minor arc estimates sufficiently to unconditionally prove the weak Goldbach conjecture. Here, the major arcs formula_3 is the union of intervals formula_4 around the rationals formula_5 where formula_6 is a constant. Minor arcs formula_7 are defined to be formula_8.