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.)

This conjecture is called "weak" because if 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 found proof of Goldbach's weak conjecture; previous results had already shown it to be true for all odd numbers greater than $e^{3100}\approx 2 \times 10^{1346}$. This last result still needs to be verified.

Some state the conjecture as:

Every odd number greater than 7 can be expressed as the sum of three odd primes.^[1]

This version excludes 7 = 2+2+3 because this requires the even prime 2. Helfgott's claim covers both versions of the conjecture.

Timeline of results

In 1923, Hardy and 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.^[2] This number has 6,846,169 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^[3] 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.^[4]

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.^[5] In 2012, Terence Tao proved this without the Riemann Hypothesis; this improves both results.^[6]

In 2002, Liu Ming-Chit (University of Hong Kong) and Wang Tian-Ze lowered this threshold to approximately $n>e^{3100}\approx 2 \times 10^{1346}$. 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.^[7][8][9] Here, the major arcs $\mathfrak M$ is the union of intervals $\left (a/q-cr_0/qx,a/q+cr_0/qx\right )$ around the rationals $a/q,q<r_0$ where $c$ is a constant. Minor arcs $\mathfrak{m}$ are defined to be $\mathfrak{m}=(\mathbb R/\mathbb Z)\setminus\mathfrak{M}$.

References

1. Weisstein, Eric W., "Goldbach Conjecture", MathWorld.
2. Golomb gives the date of Borozdin's proof as 1956; in contrast, Tao states that it was "soon after" Vinogradov's 1937 proof. Golomb, Solomon W. (1985), "The invincible primes", The Sciences 25 (2): 50–57, doi:10.1002/j.2326-1951.1985.tb02782.x; Tao, Terence (2011), "Structure and Randomness in the Prime Numbers", in Schleicher, Dierk; Lackmann, Malte, An Invitation to Mathematics: From Competitions to Research, Springer, pp. 1–7, doi:10.1007/978-3-642-19533-4_1, footnote 7, p. 1.
3. 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.
4. 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.
5. Kaniecki, Leszek (1995). "On Šnirelman's constant under the Riemann hypothesis" (PDF). Acta Arithmetica 72: 361–374. MR 1348203.
6. Tao, Terence (2014). "Every odd number greater than 1 is the sum of at most five primes". Math. Comp. 83 (286): 997–1038. arXiv:1201.6656v4. doi:10.1090/S0025-5718-2013-02733-0. MR 3143702.
7. Helfgott, Harald A. (2013). "Major arcs for Goldbach's theorem". arXiv:1305.2897 [math.NT].
8. Helfgott, Harald A. (2012). "Minor arcs for Goldbach's problem". arXiv:1205.5252 [math.NT].
9. Helfgott, Harald A. (2013). "The ternary Goldbach conjecture is true". arXiv:1312.7748 [math.NT].