648 SCIENCE PROGRESS 



Hermite, Smith, and Minkowski. The author, however, finds it convenient 

 in this volume to consider only special, though very important, cases of these 

 quadratic forms, such as, for example, the solution of 



^i' + ^i + . . . x\ =m 



for different values of n, or, again, of 



x'^ -\- y^ -{- z^ = fi, etc. 



When / is a linear function we have the theory of partitions, a subject 

 noteworthy for the part played both in the past and the present by English 

 mathematicians, among whom may be mentioned Cayley, Sylvester, Glaisher, 

 and MacMahon. There are also developments by Gauss, Eisenstein, Dirichlet, 

 Smith, Frobenius, and Minkowski, which have been of the greatest importance 

 in the development of the arithmetical theory of the general quadratic form, 

 and of the arithmetical properties of algebraic numbers. 



Another very important section arises from Fermat's Last Theorem, 

 concerning the impossibihty of 



xn ^ yn = ^n 



in rational numbers when w is a positive integer greater than 2. 



Many an unsophisticated writer, taking an equation of this type, say 



■*' + J'^ = ^^ 

 that is 



{x + y) {x -{- py) {X + p^y) = 2^, 



where p is a complex cube root of unity, argues at once, by analogy from 

 elementary arithmetic, that x + py, for example, is the cube of a similar 

 expression. The simple example 



(2 + V^l) (2 - V^) = 3' 



where 2 ± "V/ — 5 ^re certainly not the squares of, nor have, for a common 

 factor, expressions of the form a + & a/ — 5. where a and b are integers, 

 may show that the properties of ordinary integers cannot be applied to 

 algebraic numbers without investigation. Indeed, it has been the great 

 merit of Fermat's Last Theorem that it has directed the attention of 

 mathematicians to the study of the arithmetical properties of algebraic 

 numbers. In this way, Kummer found that the equation was impossible 

 if n was an odd prime not dividing any of the numerators of the first ^ (w — 3) 

 Bernoullian numbers. From some of his results, it has been shown in recent 

 years that if n is a prime and the equation has solutions for which all the 

 unknowns are prime to w, then ^'» — i — i is divisible by «* for 



? = 2, 3, 5, II, 17, and also for 



q = 7, 13, ig, if n is of the form 6 AT^ + 5. 



Since the prize of 100,000 marks was established in igo8 for a proof of 

 Fermat's Last Theorem, it has become very widely known, about 3,000 

 efforts, mostly futile, being published within three years of the announcement 

 of the prize. Much labour and paper would be saved if all future aspirants 

 consulted Prof. Dickson's volume ; but this is too much to hope for. 



There is also Waring's Problem, dating from 1770 and proved by Hilbert 

 in 1909, that for a given positive integer n and for any positive integer m, 

 an integer r can be found independent of m, so that 



^1" + x^^ . . . Xy** = m 



can be solved in positive integers ; and, to mention only one other, the 

 question of finding asymptotic formulae not only for the number of solutions 

 of the equation (i), but also of finding the total number corresponding to 

 m *= I, 2 . . . N. 



