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=== Partial results ===
The strong Goldbach conjecture is much more difficult than the [[weak Goldbach conjecture]], which says that every integer (equivalently, 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>
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>
Cully-Hugill and Dudek prove<ref>Michaela Cully-Hugill and Adrian W. Dudek, [[arxiv:2206.00433|An explicit mean-value estimate for the PNT in intervals]]</ref> a (partial and conditional) result on the Riemann hypothesis: there exists a sum of two odd primes in the interval <math>(x, x + 9696 \log^2 x2x]</math> for all x ≥ 2.
=== In popular culture ===
* According to [[Bertrand's postulate]], for every integer <math>n > 1</math>, there is always at least one prime <math>p</math> such that <math>n < p < 2n.</math> If the postulate were false, there would exist some integer <math>n</math> for which no prime numbers lie between <math>n</math> and <math>2n</math>, making it impossible to express <math>2n</math> as a sum of two primes.
Goldbach's conjecture is used when studying computation complexity.<ref>{{cite web |url=https://www.quantamagazine.org/how-the-slowest-computer-programs-illuminate-maths-fundamental-limits-20201210/ |title=How the Slowest Computer Programs Illuminate Math’sMath's Fundamental Limits|date=10 December 2020 }}</ref> The connection is made through the [[Busy Beaver]] function, where BB(''n'') is the maximum number of steps taken by any ''n'' state [[Turing machine]] that halts. There is a 27 state Turing machine that halts if and only if Goldbach's conjecture is false. Hence if BB(27) was known, and the GoldbachTuring machine did not stop in that number of steps, it would be known to run forever and hence no counterexamples exist (which proves the conjecture true). This is a completely impractical way to settle the conjecture; instead it is used to suggest that BB(27) will be very hard to compute, at least as difficult as settling the Goldbach conjecture.
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