### 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.*

**strong Goldbach Conjecture**.

The **weak Goldbach Conjecture** is:

*Every odd number greater than*7

*is the sum of three odd primes.*

^{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 |