VI.21 Edward Waring

b. Shrewsbury, England, ca. 1735; d. Shrewsbury, 1798

Lucasian Professor of Mathematics, Cambridge (1760-98)

Waring, the leading British mathematician of the latter half of the eighteenth century, wrote several advanced but somewhat impenetrable analytical texts. His first work, Miscellanea Analytica (1762), is devoted to the theory of numbers and algebraic equations and contains many results which he revised and expanded in his Meditationes Algebraicae (1770). Included in the latter is the problem known today as Waring’s problem (that every positive integer is the sum of not more than nine cubes, or the sum of not more than nineteen fourth powers, and so on, with a fixed number of summands depending on the exponent), which was solved by HILBERT [VI.63] in the affirmative in 1909 and which gave rise to important work by HARDY [VI.73] and LITTLEWOOD [VI.79] in the 1920s. The Meditationes also contained the first publication of Goldbach’s conjecture (that every even integer greater than 2 can be written as the sum of two primes) and of Wilson’s theorem (if p is a prime number, then (p — 1)! + 1 is divisible by p), which was subsequently proved by LAGRANGE [VI.22].

Waring’s problem and Goldbach’s conjecture are discussed in PROBLEMS AND RESULTS IN ADDITIVE NUMBER THEORY [V.27].