Goldbach's weak conjecture

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Goldbach's weak conjecture

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

Everyodd numbergreater than 5 can be expressed as the sum of threeprimes. (A prime may be used more than once in the same sum.)

This conjecture is called "weak" because if Goldbach's strong conjecture (concerning sums of two primes) is proven, it would be true. For if every even number greater than 4 is the sum of two odd primes, adding 3 to each even number greater than 4 will produce the odd numbers greater than 7 (and 7 itself is equal to 2+2+3).

In 2013, Harald Helfgott published a proof of Goldbach's weak conjecture.^[1] As of 2018, the proof is widely accepted in the mathematics community,^[2] but it has not yet been published in a peer-reviewed journal.

Some state the conjecture as

Every odd number greater than 7 can be expressed as the sum of three odd primes.^[3]

This version excludes 7 = 2+2+3 because this requires the even prime 2. On odd numbers larger than 7 it is slightly stronger as it also excludes sums like 17 = 2+2+13, which are allowed in the other formulation. Helfgott's proof covers both versions of the conjecture. Like the other formulation, this one also immediately follows from Goldbach's strong conjecture.

Timeline of results

In 1923,HardyandLittlewoodshowed that, assuming thegeneralized Riemann hypothesis, the weak Goldbach conjecture is true for allsufficiently largeodd numbers. In 1937,Ivan Matveevich Vinogradoveliminated the dependency on the generalised Riemann hypothesis and proved directly (seeVinogradov's theorem) that allsufficiently largeodd numbers can be expressed as the sum of three primes. Vinogradov's original proof, as it used the ineffectiveSiegel–Walfisz theorem, did not give a bound for "sufficiently large"; his student K. Borozdkin (1956) derived thatis large enough.^[4] The integer part of this number has 4,008,660 decimal digits, so checking every number under this figure would be completely infeasible.

In 1997, Deshouillers, Effinger, te Riele and Zinoviev published a result showing^[5] 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.^[6]

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.^[7] In 2012, Terence Tao proved this without the Riemann Hypothesis; this improves both results.^[8]

In 2002, Liu Ming-Chit (University of Hong Kong) and Wang Tian-Ze lowered Borozdkin's threshold to approximately. Theexponentis still much too large to admit checking all smaller numbers by computer. (Computer searches have only reached as far as 1018for the strong Goldbach conjecture, and not much further than that for the weak Goldbach conjecture.)

In 2012 and 2013, Peruvian mathematicianHarald Helfgottreleased a pair of papers improvingmajor and minor arcestimates sufficiently to unconditionally prove the weak Goldbach conjecture.^[9]^[10]^[1]^[11] Here, the major arcsis the union of intervalsaround the rationalswhereis a constant. Minor arcsare defined to be.

References

[1]

Citation Link//arxiv.org/archive/math.NTHelfgott, Harald A. (2013). "The ternary Goldbach conjecture is true". arXiv:1312.7748 [math.NT].

Sep 19, 2019, 7:57 AM

[2]

Citation Linkwww.humboldt-professur.de"Alexander von Humboldt-Professur - Harald Andrés Helfgott". www.humboldt-professur.de. Retrieved 2018-06-17.

Sep 19, 2019, 7:57 AM

[3]

Citation Linkmathworld.wolfram.comWeisstein, Eric W. "Goldbach Conjecture". MathWorld.

Sep 19, 2019, 7:57 AM

[4]

Citation Link//arxiv.org/archive/math.NTHelfgott, Harald Andrés (2015). "The ternary Goldbach problem". arXiv:1501.05438 [math.NT].

Sep 19, 2019, 7:57 AM

[5]

Citation Link//www.ams.org/mathscinet-getitem?mr=1469323Deshouillers, Jean-Marc; Effinger, Gove W.; Te Riele, Herman J. J.; Zinoviev, Dmitrii (1997). "A complete Vinogradov 3-primes theorem under the Riemann hypothesis". Electronic Research Announcements of the American Mathematical Society. 3 (15): 99–104. doi:10.1090/S1079-6762-97-00031-0. MR 1469323.

Sep 19, 2019, 7:57 AM

[6]

Citation Link//www.ams.org/mathscinet-getitem?mr=1451327Yannick Saouter (1998). "Checking the odd Goldbach Conjecture up to 1020" (PDF). Math. Comp. 67 (222): 863–866. doi:10.1090/S0025-5718-98-00928-4. MR 1451327.

Sep 19, 2019, 7:57 AM

[7]

Citation Link//www.ams.org/mathscinet-getitem?mr=1348203Kaniecki, Leszek (1995). "On Šnirelman's constant under the Riemann hypothesis" (PDF). Acta Arithmetica. 72 (4): 361–374. doi:10.4064/aa-72-4-361-374. MR 1348203.

Sep 19, 2019, 7:57 AM

[8]

Citation Link//www.ams.org/mathscinet-getitem?mr=3143702Tao, Terence (2014). "Every odd number greater than 1 is the sum of at most five primes". Math. Comp. 83 (286): 997–1038. arXiv:1201.6656. doi:10.1090/S0025-5718-2013-02733-0. MR 3143702.

Sep 19, 2019, 7:57 AM

[9]

Citation Link//arxiv.org/archive/math.NTHelfgott, Harald A. (2013). "Major arcs for Goldbach's theorem". arXiv:1305.2897 [math.NT].

Sep 19, 2019, 7:57 AM

[10]

Citation Link//arxiv.org/archive/math.NTHelfgott, Harald A. (2012). "Minor arcs for Goldbach's problem". arXiv:1205.5252 [math.NT].

Sep 19, 2019, 7:57 AM

[11]

Citation Link//arxiv.org/archive/math.NTHelfgott, Harald A. (2015). "The ternary Goldbach problem". arXiv:1501.05438 [math.NT].

Sep 19, 2019, 7:57 AM

[12]

Citation Linkarxiv.org1312.7748