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there were different German states, e.g. Friedrich Wöhler was a German chemist from Landgraviate of Hesse-Kassel and Wikipedia of course introduces him as German; and SPECIFICALLY Goldbach was a Königsberger mathematician, wasn't he? Tags: Manual revert Reverted |
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=== Origins ===
On 7 June 1742, the
{{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.}}
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=== Partial results ===
The strong Goldbach conjecture is much more difficult than the [[weak Goldbach conjecture]], which says that
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>
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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,
=== In popular culture ===
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* The Goldbach conjecture for [[practical number]]s, a prime-like sequence of integers, was stated by Margenstern in 1984,<ref>{{cite journal |first=M. |last=Margenstern |title=Results and conjectures about practical numbers |journal= [[Comptes rendus de l'Académie des Sciences]]|volume=299 |year=1984 |pages=895–898 }}</ref> and proved by [[Giuseppe Melfi|Melfi]] in 1996:<ref>{{cite journal |first=G. |last=Melfi |title=On two conjectures about practical numbers |journal= [[Journal of Number Theory]] |volume=56|year=1996 | pages=205–210 |doi=10.1006/jnth.1996.0012|doi-access=free }}</ref> every even number is a sum of two practical numbers.
* [[Harvey Dubner]] proposed a strengthening of the Goldbach conjecture that states that every even integer greater than 4208 is the sum of two [[twin prime]]s (not necessarily belonging to the same pair).<ref>{{Cite web|url=https://oeis.org/A007534/a007534.pdf|title=TWIN PRIME CONJECTURES|website=oeis.org}}</ref>{{better source|reason= This is raw html, although it seems to have been published in Recreational Mathematics|date=September 2023}} Only 34 even integers less than 4208 are not the sum of two twin primes; Dubner has verified computationally that this list is complete up to <math>2\cdot 10^{10}.</math><ref>{{Cite OEIS|A007534|name=Even numbers that are not the sum of a pair of twin primes}}</ref>{{check|reason=This source states 10^9, and the other source is unclear on the limit|date=September 2023}} A proof of this stronger conjecture would not only imply Goldbach's conjecture, but also the [[twin prime conjecture]].
* 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
{{clear}}
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