Editing Goldbach's conjecture (section)

=== 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.}}