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In [[number theory]], '''Goldbach's weak conjecture''', also known as the '''odd Goldbach conjecture''', the '''ternary Goldbach problem''', or the '''3-primes problem''', states that:

: Every [[odd number]] greater than 5 can be expressed as the sum of three [[prime number|primes]]. (A prime may be used more than once in the same sum.)

This [[conjecture]] is called "weak" because if [[Goldbach's conjecture|Goldbach's strong conjecture]] (concerning sums of two primes) is proven, it would be true. (Since if every even number greater than 4 is the sum of two odd primes, merely adding 3 to each even number greater than 4 will produce the odd numbers greater than 7.) In 2013, [[Harald Helfgott]] claimed to have fully proved the conjecture for all odd integers greater than 5 (rather than the much larger <math>e^{3100}\approx 2 \times 10^{1346}</math>, implied by previous results).

Some state the conjecture as:
:Every odd number greater than 7 can be expressed as the sum of three odd primes.<ref>{{MathWorld|title=Goldbach Conjecture|id=GoldbachConjecture}}</ref>
This version excludes 7 = 2+2+3 because this requires the even prime 2. Helfgott's claim covers both versions of the conjecture.

== Status ==
Earlier partial and/or conditional results on the conjecture include the following: In 1923, [[G. H. Hardy|Hardy]] and [[John Edensor Littlewood|Littlewood]] showed that, assuming the [[generalized Riemann hypothesis]], the odd Goldbach conjecture is true for all sufficiently large odd numbers. In 1937, [[Ivan Matveevich Vinogradov]] eliminated the dependency on the generalised Riemann hypothesis and proved directly (see [[Vinogradov's theorem]]) that all [[sufficiently large]] odd numbers can be expressed as the sum of three primes. Vinogradov's original proof, as it used the ineffective [[Siegel–Walfisz theorem]], did not give a bound for "sufficiently large"; his student K. Borozdin proved that 3315 is large enough.<ref>Golomb gives the date of Borozdin's proof as 1956; in contrast, Tao states that it was "soon after" Vinogradov's 1937 proof. {{citation|title=The invincible primes|first=Solomon W.|last=Golomb|authorlink=Solomon W. Golomb|journal=The Sciences|volume=25|issue=2|pages=50–57|year=1985|doi=10.1002/j.2326-1951.1985.tb02782.x}}; {{citation|doi=10.1007/978-3-642-19533-4_1|contribution=Structure and Randomness in the Prime Numbers|first=Terence|last=Tao|authorlink=Terence Tao|pages=1–7|year=2011|title=An Invitation to Mathematics: From Competitions to Research|editor1-first=Dierk|editor1-last=Schleicher|editor2-first=Malte|editor2-last=Lackmann|publisher=Springer}}, footnote 7, [http://books.google.com/books?id=9TATfteVeVYC&pg=PA1 p. 1].</ref> This number has 6,846,169 decimal digits, so checking every number under this figure would be completely unfeasible.

In 1997, [[Jean-Marc Deshouillers|Deshouillers]], Effinger, [[Herman te Riele|te Riele]] and Zinoviev published a result showing<ref>{{cite journal|title=A complete Vinogradov 3-primes theorem under the Riemann hypothesis|author=Deshouillers, Effinger, Te Riele and Zinoviev|journal=Electronic Research Announcements of the American Mathematical Society|volume=3|pages=99–104|year=1997|url=http://www.ams.org/era/1997-03-15/S1079-6762-97-00031-0/S1079-6762-97-00031-0.pdf|format=PDF|doi=10.1090/S1079-6762-97-00031-0|issue=15}}</ref> that the [[generalized Riemann hypothesis]] implies Goldbach's weak conjecture for all numbers. This result combines a general statement valid for numbers greater than 1020 with an extensive computer search of the small cases. Saouter also conducted a computer search covering the same cases at approximately the same time.<ref>{{cite journal|title=Checking the odd Goldbach Conjecture up to 1020|author=Yannick Saouter|journal=Mathematics of Computation|volume=67|pages=863–866|year=1998|url=http://www.ams.org/journals/mcom/1998-67-222/S0025-5718-98-00928-4/S0025-5718-98-00928-4.pdf|format=PDF|doi=10.1090/S0025-5718-98-00928-4 |issue=222}}</ref>

[[Olivier Ramaré]] in 1995 showed that every even number ''n'' ≥ 4 is in fact the sum of at most six primes, from which it follows that every odd number ''n'' ≥ 5 is the sum of at most seven primes. [[Leszek Kaniecki]] showed every odd integer is a sum of at most five primes, under the [[Riemann Hypothesis]].<ref>{{cite journal|title=On Šnirelman's constant under the Riemann hypothesis|last=Kaniecki|first=Leszek|journal=Acta Arithmetica|volume=72|year=1995|pages=361–374|url=http://matwbn.icm.edu.pl/ksiazki/aa/aa72/aa7246.pdf}}</ref> In 2012, [[Terence Tao]] proved this without the Riemann Hypothesis; this improves both results.<ref>{{Cite arXiv|last=Tao |first=Terence|title=Every odd number greater than 1 is the sum of at most five primes |eprint=1201.6656v4 |year=2012 |bibcode=2012arXiv1201.6656T |class=math.NT}}</ref>

In 2002, Liu Ming-Chit ([[University of Hong Kong]]) and Wang Tian-Ze lowered this threshold to approximately <math>n>e^{3100}\approx 2 \times 10^{1346}</math>. The [[exponent]] is still much too large to admit checking all smaller numbers by computer. (Computer searches have only reached as far as 1018 for the strong Goldbach conjecture, and not much further than that for the weak Goldbach conjecture.)

In 2012 and 2013, Peruvian mathematician [[Harald Helfgott]] released a pair of papers claiming to improve major and minor arc estimates sufficiently to unconditionally prove the weak Goldbach conjecture.<ref>{{cite arXiv |eprint=1305.2897 |title = Major arcs for Goldbach's theorem|last = Helfgott|first = H.A. |class=math.NT |year=2013}}</ref><ref> {{cite arXiv |eprint=1205.5252/ |title = Minor arcs for Goldbach's problem |last = Helfgott|first = H.A.|class=math.NT |year=2012}}</ref><ref>{{cite arXiv |eprint=1312.7748 |title = The ternary Goldbach conjecture is true|last = Helfgott|first = H.A. |class=math.NT |year=2013}}</ref> Here, the major arcs <math>\mathfrak M</math> is the union of intervals <math>\left (a/q-cr_0/qx,a/q+cr_0/qx\right )</math> around the rationals <math>a/q,q<r_0</math> where <math>c</math> is a contant. Minor arcs <math>\mathfrak{m}</math> are defined to be <math>\mathfrak{m}=(\mathbb R/\mathbb Z)\setminus\mathfrak{M}</math>.

==References==
{{reflist}}

[[Category:Additive number theory]]
[[Category:Analytic number theory]]
[[Category:Conjectures about prime numbers]]

[[ru:Проблема Гольдбаха#Тернарная проблема Гольдбаха]]

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