Editing Goldbach's conjecture (section)

==History==

=== Origins ===
On 7 June 1742, the [[Prussia]]n mathematician [[Christian Goldbach]] wrote a letter to [[Leonhard Euler]] (letter XLIII),<ref>{{cite web|url=http://eulerarchive.maa.org/correspondence/letters/OO0765|title=Letter XLIII, Goldbach to Euler|work=Correspondence of Leonhard Euler|publisher=Mathematical Association of America|date=7 June 1742|access-date=2025-01-19}}</ref> in which he proposed the following conjecture:
{{block indent|text={{lang|de|dass jede Zahl, welche aus zweyen numeris primis zusammengesetzt ist, ein aggregatum so vieler numerorum primorum sey, als man will (die unitatem mit dazu gerechnet), bis auf die congeriem omnium unitatum}}<br />
Every integer that can be written as the sum of two primes can also be written as the sum of as many primes as one wishes, until all terms are units.}}
Goldbach was following the now-abandoned convention of [[Prime number#Primality of one|considering 1]] to be a [[prime number]],<ref name=MathWorldConj>{{MathWorld|title=Goldbach Conjecture|urlname=GoldbachConjecture}}</ref> so that a sum of units would be a sum of primes.
He then proposed a second conjecture in the margin of his letter, which implies the first:<ref>In the printed version published by P.&nbsp;H. Fuss [http://eulerarchive.maa.org//correspondence/letters/OO0765.pdf] 2 is misprinted as 1 in the marginal conjecture.</ref>
{{blockquote|text={{lang|de|Es scheinet wenigstens, dass eine jede Zahl, die grösser ist als 2, ein aggregatum trium numerorum primorum sey.}}<br />
It seems at least, that every integer greater than 2 can be written as the sum of three primes.}}

Euler replied in a letter dated 30 June 1742<ref>{{cite web|url=http://eulerarchive.maa.org/correspondence/letters/OO0766.pdf|title=Letter XLIV, Euler to Goldbach|work=Correspondence of Leonhard Euler|publisher=Mathematical Association of America|date=30 June 1742|access-date=2025-01-19}}</ref> and reminded Goldbach of an earlier conversation they had had ("{{lang|de|...&nbsp;so Ew vormals mit mir communicirt haben&nbsp;...}}"), in which Goldbach had remarked that the first of those two conjectures would follow from the statement
{{block indent|Every positive even integer can be written as the sum of two primes.}}
This is in fact equivalent to his second, marginal conjecture.
In the letter dated 30 June 1742, Euler stated:<ref name="theorema">{{cite web
 |last       = Ingham
 |first      = A. E.
 |title      = Popular Lectures
 |url        = http://www.claymath.org/Popular_Lectures/U_Texas/Riemann_1.pdf
 |archive-url = https://web.archive.org/web/20030616020619/http://claymath.org/Popular_Lectures/U_Texas/Riemann_1.pdf
 |url-status   = dead
 |archive-date = 2003-06-16
 |access-date = 2009-09-23
}}</ref><ref name="PrimeGlossary">{{cite web
  | last = Caldwell
 | first = Chris
 | title = Goldbach's conjecture
 | year = 2008
 | url = http://primes.utm.edu/glossary/page.php?sort=goldbachconjecture
 | access-date = 2008-08-13 }}</ref>
{{blockquote|text={{lang|de|Dass&nbsp;... ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, ungeachtet ich dasselbe nicht demonstriren kann.}}<br />That&nbsp;... every even integer is a sum of two primes, I regard as a completely certain theorem, although I cannot prove it.}}

===Similar conjecture by Descartes===
[[René Descartes]] wrote that "Every even number can be expressed as the sum of at most three primes."<ref>[https://real.mtak.hu/164172/1/PJ_DESCARTES_Conjecture1109.pdf ''On a conjecture of Descartes''] János Pintz, ELKH R´enyi Mathematical Institute of the Hungarian Academy of Sciences. Retrieved 11 January 2025.
</ref> The proposition is similar to, but weaker than, Goldbach's conjecture. [[Paul Erdős]] said that "Descartes actually discovered this before Goldbach... but it is better that the conjecture was named for Goldbach because, mathematically speaking, Descartes was infinitely rich and Goldbach was very poor."<ref>{{cite book |last=Hoffman |first=Paul |date=1998|title=The Man Who Loved Only Numbers |location= United States|publisher=Hyperion Books |page=36 |isbn=978-0786863624}}</ref>

=== 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.

=== Computational results ===
For small values of {{mvar|n}}, the strong Goldbach conjecture (and hence the weak Goldbach conjecture) can be verified directly. For instance, in 1938, Nils Pipping laboriously verified the conjecture up to {{math|''n'' {{=}} {{val|100,000}}}}.<ref>Pipping, Nils (1890–1982), "Die Goldbachsche Vermutung und der Goldbach-Vinogradowsche Satz". Acta Acad. Aboensis, Math. Phys. 11, 4–25, 1938.</ref> With the advent of computers, many more values of {{mvar|n}} have been checked; T. Oliveira e Silva ran a distributed computer search that has verified the conjecture for {{math|''n'' ≤ {{val|4e18}}}} (and double-checked up to {{val|4e17}}) as of 2013. One record from this search is that {{val|3,325,581,707,333,960,528}} is the smallest number that cannot be written as a sum of two primes where one is smaller than 9781.<ref>Tomás Oliveira e Silva, [https://sweet.ua.pt/tos/goldbach.html Goldbach conjecture verification]. Retrieved 20 April 2024.</ref>

=== In popular culture ===
''Goldbach's Conjecture'' ({{zh|t=哥德巴赫猜想}}) is the title of the biography of Chinese mathematician and number theorist [[Chen Jingrun]], written by [[Xu Chi]].

The conjecture is a central point in the plot of the 1992 novel ''[[Uncle Petros and Goldbach's Conjecture]]'' by Greek author [[Apostolos Doxiadis]], in the short story "[[Sixty Million Trillion Combinations]]" by [[Isaac Asimov]] and also in the 2008 mystery novel ''No One You Know'' by [[Michelle Richmond]].<ref>{{Cite web |title=MathFiction: No One You Know (Michelle Richmond) |url=http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf711 |website=kasmana.people.cofc.edu}}</ref>

Goldbach's conjecture is part of the plot of the 2007 Spanish film ''[[Fermat's Room]]''.

Goldbach's conjecture is featured as the main topic of research of the titular character Marguerite in the 2023 French-Swiss film ''[[Marguerite's Theorem]]''.<ref> Odile Morain [https://www.francetvinfo.fr/culture/cinema/sorties-de-films/le-theoreme-de-marguerite-jean-pierre-darroussin-et-ella-rumpf-dans-la-folie-creatrice-des-maths_6147984.html ''Le Théorème de Marguerite''], in
[[France Télévisions|franceinfo:culture]]</ref>