Editing Goldbach's weak conjecture (section)

==Timeline of results==
In 1923, [[G. H. Hardy|Hardy]] and [[John Edensor Littlewood|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 <math>e^{e^{16.038}}\approx3^{3^{15}}</math> is large enough.<ref>{{cite arXiv |eprint=1501.05438 |title = The ternary Goldbach problem|last = Helfgott|first = Harald Andrés |class=math.NT |year=2015}}</ref> 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, [[Jean-Marc Deshouillers|Deshouillers]], Effinger, [[Herman te Riele|te Riele]] and Zinoviev published a result showing<ref>{{cite journal|title=A complete Vinogradov 3-primes theorem under the Riemann hypothesis|last1=Deshouillers | first1=Jean-Marc | last2=Effinger | first2=Gove W. | last3=Te Riele | first3=Herman J. J. | first4=Dmitrii | last4=Zinoviev | mr=1469323 | doi=10.1090/S1079-6762-97-00031-0 |journal=Electronic Research Announcements of the American Mathematical Society|volume=3|pages=99–104|year=1997|issue=15| doi-access=free | url=https://ir.cwi.nl/pub/1330/1330D.pdf }}</ref> 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<sup>20</sup> with an extensive computer search of the small cases.  Saouter also conducted a computer search covering the same cases at approximately the same time.<ref>{{cite journal|title=Checking the odd Goldbach Conjecture up to 10<sup>20</sup>|author=Yannick Saouter|journal=[[Math. Comp.]]|volume=67|pages=863–866|year=1998|url=https://www.ams.org/journals/mcom/1998-67-222/S0025-5718-98-00928-4/S0025-5718-98-00928-4.pdf|doi=10.1090/S0025-5718-98-00928-4 |issue=222 | mr=1451327|doi-access=free}}</ref>

[[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]].<ref>{{cite journal|title=On Šnirelman's constant under the Riemann hypothesis|last=Kaniecki|first=Leszek|journal=[[Acta Arithmetica]]|volume=72|issue=4|year=1995|pages=361–374|url=http://matwbn.icm.edu.pl/ksiazki/aa/aa72/aa7246.pdf|mr=1348203|doi=10.4064/aa-72-4-361-374|doi-access=free}}</ref> In 2012, [[Terence Tao]] proved this without the Riemann Hypothesis; this improves both results.<ref>{{Cite journal|last=Tao |first=Terence|title=Every odd number greater than 1 is the sum of at most five primes |arxiv=1201.6656 |year=2014 | pages=997–1038 | mr=3143702 | journal=[[Math. Comp.]] | number=286 | volume=83 | doi=10.1090/S0025-5718-2013-02733-0|s2cid=2618958}}</ref>

In 2002, Liu Ming-Chit ([[University of Hong Kong]]) and Wang Tian-Ze lowered Borozdkin's threshold to approximately <math>n>e^{3100}\approx 2 \times 10^{1346}</math>. The [[exponent]] is still much too large to admit checking all smaller numbers by computer. (Computer searches have only reached as far as 10<sup>18</sup> 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 [[Hardy–Littlewood circle method|major and minor arc]] estimates sufficiently to unconditionally prove the weak Goldbach conjecture.<ref>{{cite arXiv |eprint=1205.5252 |title = Minor arcs for Goldbach's problem |last = Helfgott|first = Harald A.|class=math.NT |year=2012}}</ref><ref>{{cite arXiv |eprint=1305.2897 |title = Major arcs for Goldbach's theorem|last = Helfgott|first = Harald A. |class=math.NT |year=2013}}</ref><ref name=":0">{{cite arXiv |eprint=1312.7748 |title = The ternary Goldbach conjecture is true|last = Helfgott|first = Harald A. |class=math.NT |year=2013}}</ref><ref>{{Cite book |editor-last=Jang |editor-first=Sun Young |last=Helfgott |first=Harald Andres |date=2014 |chapter=The ternary Goldbach problem |chapter-url=https://www.imj-prg.fr/wp-content/uploads/2020/prix/helfgott2014.pdf |title=Seoul [[International Congress of Mathematicians]] Proceedings |volume=2 |publisher=Kyung Moon SA |location=Seoul, KOR |pages=391–418 |isbn=978-89-6105-805-6 |oclc=913564239 }}</ref><ref>{{cite arXiv | eprint=1501.05438| last=Helfgott | first=Harald A. | class = math.NT | year = 2015 | title=The ternary Goldbach problem}}</ref> Here, the major arcs <math>\mathfrak M</math> is the union of intervals <math>\left (a/q-cr_0/qx,a/q+cr_0/qx\right )</math> around the rationals <math>a/q,q<r_0</math> where <math>c</math> is a constant. Minor arcs <math>\mathfrak{m}</math> are defined to be <math>\mathfrak{m}=(\mathbb R/\mathbb Z)\setminus\mathfrak{M}</math>.