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

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

Goldbach's weak conjecture

Letter from Goldbach to Euler dated on 7 June 1742 (Latin-German)[1]
Field	Number theory
Conjectured by	Christian Goldbach
Conjectured in	1742
First proof by	Harald Helfgott
First proof in	2013
Implied by	Goldbach's conjecture

This conjecture is called "weak" because if Goldbach's strong conjecture (concerning sums of two primes) is proven, then this would also 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.[2] As of 2018, the proof is widely accepted in the mathematics community,[3] 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.[4]

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.

Origins

On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII),[5] in which he proposed the following conjecture:

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.

He then proposed a second conjecture in the margin of his letter:

Every integer greater than 2 can be written as the sum of three primes.

He considered 1 to be a prime number, a convention subsequently abandoned.[4] The two conjectures are now known to be equivalent, but this did not seem to be an issue at the time. A modern version of Goldbach's marginal conjecture (now known as Goldbach's weak conjecture) is:

Every integer greater than 5 can be written as the sum of three primes.

Euler replied in a letter dated 30 June 1742 and reminded Goldbach of an earlier conversation they had ("…so Ew vormals mit mir communicirt haben…"), in which Goldbach remarked his original (and not marginal) conjecture followed from the following statement

Every even integer greater than 2 can be written as the sum of two primes,

which is, thus, also a conjecture of Goldbach. In the letter dated 30 June 1742, Euler stated:[6][7]

"Dass … ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, ungeachtet ich dasselbe nicht demonstriren kann." ("That … every even integer is a sum of two primes, I regard as a completely certain theorem, although I cannot prove it.")

Timeline of results

In 1923, Hardy and Littlewood showed that, assuming the generalized Riemann hypothesis, the weak 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. Borozdkin (1956) derived that ${\displaystyle e^{e^{16.038}}\approx 3^{3^{15}}}$ is large enough.[8] 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[9] 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.[10]

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.[11] In 2012, Terence Tao proved this without the Riemann Hypothesis; this improves both results.[12]

In 2002, Liu Ming-Chit (University of Hong Kong) and Wang Tian-Ze lowered Borozdkin's threshold to approximately ${\displaystyle n>e^{3100}\approx 2\times 10^{1346}}$. 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 improving major and minor arc estimates sufficiently to unconditionally prove the weak Goldbach conjecture.[13][14][2][15] Here, the major arcs ${\displaystyle {\mathfrak {M}}}$ is the union of intervals ${\displaystyle \left(a/q-cr_{0}/qx,a/q+cr_{0}/qx\right)}$ around the rationals ${\displaystyle a/q,q<r_{0}}$ where ${\displaystyle c}$ is a constant. Minor arcs ${\displaystyle {\mathfrak {m}}}$ are defined to be ${\displaystyle {\mathfrak {m}}=(\mathbb {R} /\mathbb {Z} )\setminus {\mathfrak {M}}}$.

References

1. Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle (Band 1), St.-Pétersbourg 1843, pp. 125–129.
2. Helfgott, Harald A. (2013). "The ternary Goldbach conjecture is true". arXiv:1312.7748 [math.NT].
3. "Alexander von Humboldt-Professur - Harald Andrés Helfgott". www.humboldt-professur.de. Retrieved 2018-06-17.
4. Weisstein, Eric W. "Goldbach Conjecture". MathWorld.
5. http://www.math.dartmouth.edu/~euler/correspondence/letters/OO0765.pdf
6. Ingham, A. E. "Popular Lectures" (PDF). Archived from the original (PDF) on 2003-06-16. Retrieved 2009-09-23.
7. Caldwell, Chris (2008). "Goldbach's conjecture". Retrieved 2008-08-13.
8. Helfgott, Harald Andrés (2015). "The ternary Goldbach problem". arXiv:1501.05438 [math.NT].
9. Deshouillers, 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.
10. Yannick 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.
11. Kaniecki, 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.
12. Tao, 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.
13. Helfgott, Harald A. (2013). "Major arcs for Goldbach's theorem". arXiv:1305.2897 [math.NT].
14. Helfgott, Harald A. (2012). "Minor arcs for Goldbach's problem". arXiv:1205.5252 [math.NT].
15. Helfgott, Harald A. (2015). "The ternary Goldbach problem". arXiv:1501.05438 [math.NT].