674 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxm 



similarly from another root of the congruence. The process is simplified, 

 applied to x' 2 +qy 2 = m and compared with the method of binary quadratic 

 forms. 



CONDITIONS FOR AN INFINITUDE OF SOLUTIONS OF f(x, y) = 0. 



C. Runge 5 considered an irreducible polynomial f(x, y] with integral 

 coefficients (i. e., not a product of such polynomials), and the algebraic 

 function y defined by f(x, y] = 0. By one system of conjugate developments 

 of y according to descending powers of x is meant those obtained from a 

 single development by replacing the single algebraic number, in terms of 

 which all the coefficients are expressed rationally, by its conjugate values 

 and the fractional power of x by all its values. He proved that if the 

 various developments of y form more than one system of conjugates there 

 is only a finite number of integral values of x for which f(x, y) = is satisfied 

 by rational values of y. Also that /= has an infinitude of pairs of integral 

 solutions x, y only when x, y become infinite simultaneously and when the 

 developments according to descending powers of one of these variables 

 form a single system of conjugate developments. Hence necessary (but 

 not sufficient) conditions for an infinitude of pairs of integral solutions 

 x, y of f(x, y} = are : (i) If / is of degree m in x and n in y, the coefficients 

 of x m and y n are constants a, 6. (ii) The algebraic function y defined by 

 f(x, y) = becomes infinite with x with the order of x mln . If cx p y is a 

 term of/, then np+m<r^mn. (iii) The sum of the terms for which 



np+ma = mn 

 must be expressible in the form 



6n(t/ x -<V) 03 = 1, 2, '. ., n/X), 

 



where U(u d p ) is a power of an irreducible function of u. 



A. Boutin 6 raised the question as to the types of equations such that, 

 if Xi, yi (i = n l, n2) are two sets of integral solutions, 



(1) X n = aX n -l + PX n -z, ?/ n =a?/ n _i + /ft/n-2 



are also solutions. E. Maillet 7 treated the properties of one or two recurring 

 series x n +p = cxix n+p -i-i ----- \-a p x n with rational (or integral) coefficients and 

 proved that the only equations F(x, y} = 0, where F is without a rational 

 divisor, with an infinitude of integral solutions given by a formula of 

 recurrence (1) of the second order are either linear, quadratic 



or (tv'y t'vx) p (vu'x - uv'yY(tu' ut') p ~ q = 0, 



where p, q are relatively prime integers. If we consider rational solutions, 

 we obtain an analogous result. 



E. Maillet 8 proved theorems concerning arithmetically irreducible equa- 

 tions 

 (2) F(x, y} = <f> n (x, y)+0n-i(s, y)-\ 



6 Jour, fur Math., 100, 1887, 425-35. 



6 L'interm&liaire des math., 1, 1894, 20-21. 



'' M6m. Acad. Sc. Toulouse, (9), 7, 1895, 182-213. 



8 Comptes Rendus Paris, 128, 1899, 1383; Jour, de Math., (5), 6, 1900, 261-77. 



