Editing Goldbach's conjecture (section)

=== Partial results ===
The strong Goldbach conjecture is much more difficult than the [[weak Goldbach conjecture]], which says that every odd integer greater than 5 is the sum of three primes. Using [[Ivan_Vinogradov#Mathematical_contributions|Vinogradov's method]], [[Nikolai Chudakov]],<ref>{{Cite journal |last=Chudakov |first=Nikolai G. |year=1937 |title={{lang|ru|О проблеме Гольдбаха}} |trans-title=On the Goldbach problem |journal=[[Doklady Akademii Nauk SSSR]] |volume=17 |pages=335–338}}</ref> [[Johannes van der Corput]],<ref>{{cite journal |last=Van der Corput |first=J. G. |year=1938 |title=Sur l'hypothèse de Goldbach |url=http://www.dwc.knaw.nl/DL/publications/PU00016746.pdf |journal=Proc. Akad. Wet. Amsterdam |language=fr |volume=41 |pages=76–80}}</ref> and [[Theodor Estermann]]<ref>{{cite journal |last=Estermann |first=T. |year=1938 |title=On Goldbach's problem: proof that almost all even positive integers are sums of two primes |journal=Proc. London Math. Soc. |series=2 |volume=44 |pages=307–314 |doi=10.1112/plms/s2-44.4.307}}</ref> showed (1937–1938) that [[almost all]] even numbers can be written as the sum of two primes (in the sense that the fraction of even numbers up to some {{mvar|N}} which can be so written tends towards 1 as {{mvar|N}} increases). In 1930, [[Lev Schnirelmann]] proved that any [[natural number]] greater than 1 can be written as the sum of not more than {{mvar|C}} prime numbers, where {{mvar|C}} is an effectively computable constant; see [[Schnirelmann density]].<ref>Schnirelmann, L. G. (1930). "[http://mi.mathnet.ru/eng/umn/y1939/i6/p9 On the additive properties of numbers]", first published in "Proceedings of the Don Polytechnic Institute in Novocherkassk" (in Russian), vol '''14''' (1930), pp. 3–27, and reprinted in "Uspekhi Matematicheskikh Nauk" (in Russian), 1939, no. 6, 9–25.</ref><ref>Schnirelmann, L. G. (1933). First published as "[https://link.springer.com/article/10.1007/BF01448914 Über additive Eigenschaften von Zahlen]" in "[[Mathematische Annalen]]" (in German), vol. '''107''' (1933), 649–690, and reprinted as "[http://mi.mathnet.ru/eng/umn/y1940/i7/p7 On the additive properties of numbers]" in "Uspekhi Matematicheskikh Nauk" (in Russian), 1940, no. 7, 7–46.</ref> Schnirelmann's constant is the lowest number {{mvar|C}} with this property. Schnirelmann himself obtained {{math|''C'' < {{val|800,000}}}}. This result was subsequently enhanced by many authors, such as [[Olivier Ramaré]], who in 1995 showed that every even number {{math|''n'' ≥ 4}} is in fact the sum of at most 6 primes. The best known result currently stems from the proof of the weak Goldbach conjecture by [[Harald Helfgott]],<ref>{{cite arXiv |eprint=1312.7748 |class=math.NT |first=H. A. |last=Helfgott |title=The ternary Goldbach conjecture is true |date=2013}}</ref> which directly implies that every even number {{math|''n'' ≥ 4}} is the sum of at most 4 primes.<ref>{{Cite journal |last=Sinisalo |first=Matti K. |date=Oct 1993 |title=Checking the Goldbach Conjecture up to 4 ⋅ 10<sup>11</sup> |url=https://www.ams.org/journals/mcom/1993-61-204/S0025-5718-1993-1185250-6/S0025-5718-1993-1185250-6.pdf |publisher=American Mathematical Society |volume=61 |issue=204 |pages=931–934 |citeseerx=10.1.1.364.3111 |doi=10.2307/2153264 |jstor=2153264 |periodical=Mathematics of Computation}}</ref><ref>{{cite book |last=Rassias |first=M. Th. |title=Goldbach's Problem: Selected Topics |publisher=Springer |year=2017}}</ref>

In 1924, Hardy and Littlewood showed under the assumption of the [[generalized Riemann hypothesis]] that the number of even numbers up to {{mvar|X}} violating the Goldbach conjecture is [[Inequality (mathematics)|much less than]] {{math|''X''<sup>{{1/2}} + ''c''</sup>}} for small {{mvar|c}}.<ref>See, for example, ''A new explicit formula in the additive theory of primes with applications I. The explicit formula for the Goldbach and Generalized Twin Prime Problems'' by Janos Pintz.</ref>

In 1948, using [[sieve theory]] methods, [[Alfréd Rényi]] showed that every sufficiently large even number can be written as the sum of a prime and an [[almost prime]] with at most {{mvar|K}} factors.<ref name="Alfréd Rényi 1948">{{cite journal |last=Rényi |first=A. A. |year=1948 |title=On the representation of an even number as the sum of a prime and an almost prime |journal=Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya |language=Russian |volume=12 |pages=57–78}}</ref> [[Chen Jingrun]] showed in 1973 using sieve theory that every [[sufficiently large]] even number can be written as the sum of either two primes, or a prime and a [[semiprime]] (the product of two primes).<ref>{{cite journal |last=Chen |first=J. R. |year=1973 |title=On the representation of a larger even integer as the sum of a prime and the product of at most two primes |journal=Sci. Sinica |volume=16 |pages=157–176}}</ref> See [[Chen's theorem]] for further information.

In 1975, [[Hugh Lowell Montgomery]] and [[Bob Vaughan]] showed that "most" even numbers are expressible as the sum of two primes. More precisely, they showed that there exist positive constants {{mvar|c}} and {{mvar|C}} such that for all sufficiently large numbers {{mvar|N}}, every even number less than {{mvar|N}} is the sum of two primes, with at most {{math|''CN''<sup>1 − ''c''</sup>}} exceptions. In particular, the set of even integers that are not the sum of two primes has [[natural density|density]] zero.

In 1951, [[Yuri Linnik]] proved the existence of a constant {{mvar|K}} such that every sufficiently large even number is the sum of two primes and at most {{mvar|K}} powers of 2. [[János Pintz]] and [[Imre Z. Ruzsa|Imre Ruzsa]] found in 2020 that {{math|1=''K'' = 8}} works.<ref>{{Cite journal |last1=Pintz |first1=J. |last2=Ruzsa |first2=I. Z. |date=2020-08-01 |title=On Linnik's approximation to Goldbach's problem. II |url=https://doi.org/10.1007/s10474-020-01077-8 |journal=[[Acta Mathematica Hungarica]] |language=en |volume=161 |issue=2 |pages=569–582 |doi=10.1007/s10474-020-01077-8 |s2cid=54613256 |issn=1588-2632 |authorlink1=János Pintz}}</ref> Assuming the [[generalized Riemann hypothesis]], {{math|1=''K'' = 7}} also works, as shown by [[Roger Heath-Brown]] and [[Jan-Christoph Schlage-Puchta]] in 2002.<ref>{{cite journal |last1=Heath-Brown |first1=D. R. |last2=Puchta |first2=J. C. |year=2002 |title=Integers represented as a sum of primes and powers of two |journal=[[Asian Journal of Mathematics]] |volume=6 |issue=3 |pages=535–565 |arxiv=math.NT/0201299 |bibcode=2002math......1299H |doi=10.4310/AJM.2002.v6.n3.a7 |s2cid=2843509}}</ref>

A proof for the weak conjecture was submitted in 2013 by [[Harald Helfgott]] to ''[[Annals of Mathematics Studies]]'' series. Although the article was accepted, Helfgott decided to undertake the major modifications suggested by the referee. Despite several revisions, Helfgott's proof has not yet appeared in a peer-reviewed publication.<ref name="Helfgott 2013">{{cite arXiv |eprint=1305.2897 |class=math.NT |first=H. A. |last=Helfgott |title=Major arcs for Goldbach's theorem |year=2013}}</ref><ref name="Helfgott 2012">{{cite arXiv |eprint=1205.5252 |class=math.NT |first=H. A. |last=Helfgott |title=Minor arcs for Goldbach's problem |year=2012}}</ref><ref>{{Cite web |title=Harald Andrés Helfgott |url=https://webusers.imj-prg.fr/~harald.helfgott/anglais/book.html |access-date=2021-04-06 |publisher=Institut de Mathématiques de Jussieu-Paris Rive Gauche}}</ref> The weak conjecture is implied by the strong conjecture, as if {{math|''n'' − 3}} is a sum of two primes, then {{mvar|n}} is a sum of three primes. However, the converse implication and thus the strong Goldbach conjecture would remain unproven if Helfgott's proof is correct.