Waring's Problem and the Goldbach Conjecture
We look here at some of the results about Waring's Problem and the Goldbach Conjecture which have been proved since Hardy gave his inaugural lecture at the University of Oxford in 1920.
1. Waring's Problem g(k).
The number g(k) is the least number such that every number is the sum of g(k) or less k-th powers.
In his 1920 inaugural lecture, Hardy knew that g(1) = 1, g(2) = 4 and g(3) = 9. He did not have an exact value for g(k) for k ≥ 4 but he gives bounds. The following has been proved since 1920:
g(4) = 19 was proved in 1986 by Ramachandran Balasubramanian, Jean-Marc Deshouillers, and François Dress in two papers.
g(5) = 37 was proved in 1964 by Chen Jingrun.
g(6) = 73 was proved in 1940 by S S Pillai.
Here are the first values of g(k):
1, 4, 9, 19, 37, 73, 143, 279, 548, 1079, 2132, 4223, 8384, 16673, 33203, 66190, 132055, 263619, 526502, 1051899, 2102137, 4201783, 8399828, ...
It is known that g(k) = 2^{k} + [(3/2)^{k}] - 2 for all k ≤ 471,600,000 where [x] is the largest integer less than x. This was proved by J M Kubina and M C Wunderlich, in their paper "Extending Waring's conjecture to 471,600,000" in Math. Comp. 55 (1990), 815-820.
2. Waring's Problem G(k).
The number G(k) is the least number such that for every integer from a certain point onwards is the sum of G(k)or less k-th powers.
Although much progress has been made in determining g(k), there has been much less progress in determining G(k). In his 1920 inaugural lecture, Hardy knew that G(1) = 1, G(2) = 4 and 4 ≤ G(3) ≤ 8. Hardy also knew that 16 ≤ G(4) ≤ 21. The following has been proved since 1920:
G(3) ≤ 7 was proved by Y V Linnik. The result was announced in 1942 in his paper "On the representation of large numbers as sums of seven cubes" in Dokl. Akad. Nauk SSSR 35 (1942), 162. A proof is given in Linnik's paper "On the representation of large numbers as sums of seven cubes" in Mat. Sb. 12 (1943), 218-224.
G(4) = 16 was proved by Harold Davenport in 1939 in his paper "On Waring's problem for fourth powers" in Ann. of Math. 40 (1939), 731-747.
For G(k), 5 ≤ k ≤ 20, we have the following results which, as of January 2017, we believe are the best obtained so far:
k | G(k) | Proved by | Journal | Year |
5 | ≤ 17 | Vaughan & Wooley | Acta Math. | 1995 |
6 | ≤ 24 | Vaughan & Wooley | Duke Math. J. | 1994 |
7 | ≤ 33 | Vaughan & Wooley | Acta Math. | 1995 |
8 | ≤ 42 | Vaughan & Wooley | Phil. Trans. Roy. Soc. | 1993 |
9 | ≤ 50 | Vaughan & Wooley | Acta Arith. | 2000 |
10 | ≤ 59 | Vaughan & Wooley | Acta Arith. | 2000 |
11 | ≤ 67 | Vaughan & Wooley | Acta Arith. | 2000 |
12 | ≤ 76 | Vaughan & Wooley | Acta Arith. | 2000 |
13 | ≤ 84 | Vaughan & Wooley | Acta Arith. | 2000 |
14 | ≤ 92 | Vaughan & Wooley | Acta Arith. | 2000 |
15 | ≤ 100 | Vaughan & Wooley | Acta Arith. | 2000 |
16 | ≤ 109 | Vaughan & Wooley | Acta Arith. | 2000 |
17 | ≤ 117 | Vaughan & Wooley | Acta Arith. | 2000 |
18 | ≤ 125 | Vaughan & Wooley | Acta Arith. | 2000 |
19 | ≤ 134 | Vaughan & Wooley | Acta Arith. | 2000 |
20 | ≤ 142 | Vaughan & Wooley | Acta Arith. | 2000 |
To illustrate the progress towards these "up-to-date" results, we give an indication of how the bounds for G(9) have been improved since Hardy gave his 1920 lecture:
≤ | Proved by | Journal | Year |
949 | G H Hardy & J E Littlewood | Math. Z. | 1922 |
824 | R D James | Proc. London Math. Soc. | 1934 |
190 | H Heilbronn | Acta Arith. | 1936 |
101 | T Estermann | Acta Arith. | 1937 |
99 | V Narasimhamurti | J. Indian Math. Soc. | 1941 |
96 | R J Cook | Bull. London Math. Soc. | 1973 |
91 | R C Vaughan | Acta Arith. | 1977 |
90 | K Thanigasalam | Acta Arith. | 1980 |
88 | K Thanigasalam | Acta Arith. | 1982 |
87 | K Thanigasalam | Acta Arith. | 1985 |
82 | R C Vaughan | J. London Math. Soc. | 1986 |
75 | R C Vaughan | Acta Math. | 1989 |
55 | T D Wooley | Ann. of Math. | 1992 |
51 | R C Vaughan & T D Wooley | Acta Math. | 1995 |
50 | R C Vaughan & T D Wooley | Acta Arith. | 2000 |
It has been shown that the following lower bounds hold
k | G(k) |
5 | ≥ 6 |
6 | ≥ 9 |
7 | ≥ 8 |
8 | ≥ 32 |
9 | ≥ 13 |
10 | ≥ 12 |
11 | ≥ 12 |
12 | ≥ 16 |
13 | ≥ 14 |
14 | ≥ 15 |
15 | ≥ 16 |
16 | ≥ 64 |
17 | ≥ 18 |
18 | ≥ 27 |
19 | ≥ 20 |
20 | ≥ 25 |
It has been conjectured that these lower bounds are the correct values for G(k).
3. Goldbach Conjecture.
Hardy states the Goldbach Conjecture in his 1920 inaugural lecture as:
Every even number greater than 2 is the sum of two odd primes.
This is sometimes today called the strong Goldbach Conjecture.
The weak Goldbach Conjecture is:
Every odd number greater than 7 is the sum of three odd primes.
In 2013, Harald Helfgott proved Goldbach's weak conjecture; previous results had already shown it to be true for all odd numbers greater than about 2 × 10^{1346}.
The strong Goldbach conjecture has been shown to hold for all n up to 4 × 10^{18}. The following table shows the progress towards this:
10^{5} | N Pipping | 1938 |
10^{8} | M L Stein & P R Stein | 1965 |
2 × 10^{10} | A Granville, J van der Lune & H J J te Riele | 1989 |
4 × 10^{11} | M K Sinisalo | 1993 |
10^{14} | J M Deshouillers, H J J te Riele & Y Saouter | 1998 |
4 × 10^{14} | J Richstein | 2001 |
2 × 10^{16} | T Oliveira e Silva | 2003 |
6 × 10^{16} | T Oliveira e Silva | 2003 |
2 × 10^{17} | T Oliveira e Silva | 2005 |
3 × 10^{17} | T Oliveira e Silva | 2005 |
12 × 10^{17} | T Oliveira e Silva | 2008 |
4 × 10^{18} | T Oliveira e Silva | 2012 |