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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:
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| : 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.)
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| 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).
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| Some state the conjecture as:
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| :Every odd number greater than 7 can be expressed as the sum of three odd primes.<ref>{{MathWorld|title=Goldbach Conjecture|id=GoldbachConjecture}}</ref>
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| This version excludes 7 = 2+2+3 because this requires the even prime 2. Helfgott's claim covers both versions of the conjecture.
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| == Status ==
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| 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 3<sup>3<sup>15</sup></sup> 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.
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| 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 10<sup>20</sup> 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 10<sup>20</sup>|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>
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| [[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>
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| 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 10<sup>18</sup> for the strong Goldbach conjecture, and not much further than that for the weak Goldbach conjecture.)
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| 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>.
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| ==References==
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| {{reflist}}
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| [[Category:Additive number theory]]
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| [[Category:Analytic number theory]]
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| [[Category:Conjectures about prime numbers]]
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| [[ru:Проблема Гольдбаха#Тернарная проблема Гольдбаха]]
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