CHAP. II] REAL LINEAR FORMS, APPROXIMATION. 93 



are solvable in integers when the /3's are arbitrary, with a desired approxima- 

 tion, if and only if Sa^a;* = (mod 1) are not solvable exactly in integers 

 not all zero. 



U. Scarpis 254 proved that a system of n linear homogeneous congruences 

 in n unknowns has solutions not all divisible by the modulus M if and only 

 if the determinant A of the coefficients is not prime to M. The problem 

 is reduced as usual to the case M = p m , where p is a prime. Then let 

 some p-rowed minor of A be prime to p, but all /b-rowed minors (k ^ p + 1) 

 be divisible by p. Let p e be the highest power of p dividing A and all its 

 Arrowed minors (k ^ p + 1). Then p of the congruences are linearly 

 independent. We may assume that | a^ , where i, j = 1, , p, is prime 

 to p. Then the last n p of the congruences can be replaced by congru- 

 ences in Xp+i, - - -, x n in which each coefficient is divisible by p e . If m = 1, 

 no more than p of the initial congruences are linearly independent; the 

 values of Xi, , x p are uniquely determined in terms of x p+ i, - , x n which 

 are arbitrary, so that there are p n ~" sets of solutions. 



LINEAR FORMS WITH REAL COEFFICENTS; APPROXIMATION. 



J. L. Lagrange 255 noted that, if a is a given positive real number, we 

 can find relatively prime positive integers p, q such that p aq shall be 

 numerically smaller than r as for r < p, s < q, by taking p/q as any 

 principal convergent to the continued fraction for a with all terms positive. 



Lagrange 22 determined a fraction m/a, with given numerator or denomi- 

 nator, which shall approximate as closely as possible to the given fraction 

 B/A < 1, where A, B are relatively prime. For example, let m be given. 

 Take as a the quotient found on dividing Am by B. If =F C is the remainder 

 numerically < \B, then Ba - Am = C, B/A = m/a C/(Aa). Start- 

 ing with C/A, determine similarly nfb, with n given, by using the quotient 

 b and remainder =F D when An is divided by C, whence C/A = n/b db D/(Ab). 

 Similarly, D/A = p/c E/(Ac). It follows that m < a, n ^ b, p ^ c, 

 - and that A, B, C, D, - form a decreasing series terminating with zero: 



B m n p 



. 



A a ab abc 



In case the denominators a, b, c, were given and all equal, we have 

 expressed B/A to the base a. Finally, suppose that neither m nor a is 

 given, but are to be found such that m < B, n < A, and such that m/a is 

 as close an approximation to B/A as possible. Hence must C = db 1. 

 Then m and a are found by Euclid's g.c.d. process. Saunderson 13 had 

 already treated the approximation to a fraction and cited earlier writers. 



C. G. J. Jacobi 256 proved that integral values not all zero can be assigned 

 to x, y, z such that ax + a'y + a"z and bx + b'y + b"z are less than any 

 assigned quantity. Cf. Sylvester 108 of Ch. III. 



254 Periodico di Mat., 23, 1908, 49-61. 



255 Additions to Euler's Algebra, 2, 1774, 445; Oeuvres, VII, 45-57. 



256 Jour, fur Math., 13, 1835, 55; Werke, II, 29-31. 



