94 HISTORY OF THE THEORY OF NUMBERS. [CHAP. II 



G. L. Dirichlet 257 stated that it has been long known from the theory of 

 continued fractions that, if a is irrational, there exists an infinitude of pairs 

 of integers x, y for which x ay is numerically < 1/y. He proved the 

 following generalization: If i, , a m are such that 



/ = X + aiXi + + a m x 



mm 



vanishes for no set of integral values of XQ, , x m , not all zero, there exists 

 an infinitude of sets of integers x , , x m , with Xi, , x m not all zero, 

 such that / is numerically < l/s m , where s is the greatest of x\, , x m . 

 Similarly for several forms /. For example, if a = aiXi + + a m x m and 

 ]8 = fiiXi + + p m x m vanish simultaneously only when Xi, , x m are 

 all zero, there exists an infinitude of sets of integers Xi, - , x m not all zero 

 for which \a < Afs a , 13 < B/s m ~ 2 ~ a , in which A and B are constants 

 depending on the at, &-, while a is any constant between and m 2. 



Ch. Hermite 258 remarked that, if A and B are given irrational numbers, 

 we can readily find the linear relations Aa + Bb + c = (if existent), 

 where a, b, c are integers. In fact, a. = mA m! and /5 = mB m" can 

 be made as small as one pleases [by choice of integers ra, m', m"~\. Since 

 aa + fc/3 = am' bm" cm is an integer, it cannot be made < 1 with- 

 out reducing to zero. Thus to find m, m f , m", we have only to convert 

 /3/a into a continued fraction to obtain the desired relation. 



Hermite 259 proved by means of the minimum of a binary quadratic form 

 that, if a, A are real, there exist integers m, n such that 



(m - an)* 



whence \m an \ < 1/(). Let m r , n' be the integers corresponding 

 to A' = A + d, where 8 is an infinitesimal. Then mn' nm' = 1. 



P. L. Tchebychef 260 proved that, if a is irrational and 6 is given, there 

 exists an infinitude of sets of integers x, y such that y ax b is numer- 

 ically < 2/| x | . 



Hermite 261 proved that, in Tcheby chef's result, we may replace 2/\ x\ 



by l/{2 | x | } and in fact by V2/27 (\x\. 



L. Kronecker 262 treated the problem to find integers w, w' such that 

 aw + a'w' takes a value as near as possible to , where a, a', % are given 

 real numbers. In general, consider a system of p equations 



anWi + " + a iq w q = & (i = 1, , p}, 



with real coefficients. Let r be the number of these equations whose left 



267 Sitzungsber. Akad. Wiss. Berlin, 1842, 93; Werke, I, 635-8. 



158 Jour, fur Math., 40, 1850, 261; Oeuvres, I, 101. 



Ibid., 41, 1851, 195-7; Oeuvres, I, 168-171. 



260 Zapieki Acad. nauk St. P6tersbourg, 10, 1866, Suppl. No. 4, p. 50; Oeuvres, 1, 1899, 679. 



M1 Jour, fur Math., 88, 1879, 10-15; Ouevres, III, 513-9. 



2 '2 Monatsber. Akad. Wiss. Berlin, 1884, 1179-93, 1271-99; Werke, III,, 47-109. Cf. ibid., 



1071-80; Comptes Rendus Paris, 96, 1883, 93-8, 148-52, 216-21; 99, 1884, 765-71, 



Werke, IIIi, 1-44, for application to algebraic units. 



