Goldbach's weak conjecture

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

Every integer greater than 5 can be written as the sum of three primes.

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).

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 ${\displaystyle e^{e^{16.038}}\approx 3^{3^{15}}}$ is large enough.[7] 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[8] that the generalized Riemann hypothesis implies Goldbach's weak conjecture for all numbers. This result combines a general statement valid for numbers greater than 10²⁰ with an extensive computer search of the small cases. Saouter also conducted a computer search covering the same cases at approximately the same time.[9]

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.[10] In 2012, Terence Tao proved this without the Riemann Hypothesis; this improves both results.[11]

In 2002, Liu Ming-Chit (University of Hong Kong) and Wang Tian-Ze lowered Borozdkin's threshold to approximately ${\displaystyle n>e^{3100}\approx 2\times 10^{1346}}$. The exponent is still much too large to admit checking all smaller numbers by computer. (Computer searches have only reached as far as 10¹⁸ 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.[12][13][2][14] Here, the major arcs ${\displaystyle {\mathfrak {M}}}$ is the union of intervals ${\displaystyle \left(a/q-cr_{0}/qx,a/q+cr_{0}/qx\right)}$ around the rationals ${\displaystyle a/q,q<r_{0}}$ where ${\displaystyle c}$ is a constant. Minor arcs ${\displaystyle {\mathfrak {m}}}$ are defined to be ${\displaystyle {\mathfrak {m}}=(\mathbb {R} /\mathbb {Z} )\setminus {\mathfrak {M}}}$.