# Waring's problem

### 1770

Waring's problem was proposed in 1770 by Edward Waring, after whom it is named.

This question later became known as Bachet's conjecture, after the 1621 translation of Diophantus by Claude Gaspard Bachet de MÃ©ziriac, and it was solved by Joseph-Louis Lagrange in his four-square theorem in 1770, the same year Waring made his conjecture.

Waring conjectured that these lower bounds were in fact exact values. Lagrange's four-square theorem of 1770 states that every natural number is the sum of at most four squares.

### 1772

Euler, the son of Leonhard Euler, in about 1772.

### 1909

Its affirmative answer, known as the Hilbertâ€“Waring theorem, was provided by Hilbert in 1909.

Hardy and Littlewood showed that all sufficiently large numbers are the sum of at most 19 fourth powers. That g(3) = 9 was established from 1909 to 1912 by Wieferich and A.

### 1912

Hardy and Littlewood showed that all sufficiently large numbers are the sum of at most 19 fourth powers. That g(3) = 9 was established from 1909 to 1912 by Wieferich and A.

### 1939

Davenport showed that in 1939, by demonstrating that any sufficiently large number congruent to 1 through 14 mod 16 could be written as a sum of 14 fourth powers (Vaughan in 1985 and 1989 reduced the 14 successively to 13 and 12).

### 1940

Deshouillers, g(5) = 37 in 1964 by Chen Jingrun, and g(6) = 73 in 1940 by Pillai. Let \lfloor x\rfloor and \{x\} respectively denote the integral and fractional part of a positive real number x.

### 1943

The upper bound is due to Linnik in 1943.

### 1964

Deshouillers, g(5) = 37 in 1964 by Chen Jingrun, and g(6) = 73 in 1940 by Pillai. Let \lfloor x\rfloor and \{x\} respectively denote the integral and fractional part of a positive real number x.

### 1985

Davenport showed that in 1939, by demonstrating that any sufficiently large number congruent to 1 through 14 mod 16 could be written as a sum of 14 fourth powers (Vaughan in 1985 and 1989 reduced the 14 successively to 13 and 12).

### 1986

Kempner, g(4) = 19 in 1986 by R.

### 1989

Davenport showed that in 1939, by demonstrating that any sufficiently large number congruent to 1 through 14 mod 16 could be written as a sum of 14 fourth powers (Vaughan in 1985 and 1989 reduced the 14 successively to 13 and 12).

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