CHAP, xxni] SINGLE EQUATIONS OF DEGREE n>4. 691 



Let f=pfq, where p, q are relatively prime. Thus Z is a multiple z l q of q 

 and z = z l p, Zi = z l qZi, z 2 = z l qZ 2 . 



A. Flechsenhaar 137 and E. Schulte discussed l/a+l/6 = l/c. E. S6s 

 (p. 113) treated (2). W. Hofmann 138 discussed the integral solutions of 



1 1 = 1 1_1 = 1_1 



a b c' abbe 



G. Lemaire 139 transformed given decompositions SI// of 9/10 into others. 

 R. Janculescu 140 noted that in l/x + l/y = Ifz, z will be integral only when 

 the g.c.d. d of x and y is a multiple of x/d+yfd. 

 D. Biddle 141 solved each of l/(a&)+l/(co) = 



MISCELLANEOUS SINGLE EQUATIONS OF DEGREE n>4. 



J. L. Lagrange 142 noted that, if a is a fixed nth root of unity, the product 

 of two functions of the type _ 



p = t+ua A/A+za 2 A/AH ----- \-za n ~ l 



is of like form. Hence if we replace a by the different nth roots of unity 

 and form the product of the functions so obtained from p, we obtain a 

 rational function P of t, u, - , z, A such that the product of two functions 

 of type P is a third function of type P. We can find P by eliminating co 

 between 



then P is the term free of I in the eliminant. For example, if n = 2, 

 P = t 2 Au 2 . An application is to the solution of 



(1) r n As n = q m . 



We seek to express each factor r asA l!n as an mth power p m , where a n = 1, 



and p is the above linear function. Then 



+Za n ~ 1 



Hence r=T, s=-U, X=0, , Z = 0. Thus (1) is solvable by this 

 method if X=Q, , Z = Q are solvable. Although only n 2 equations 

 in n variables, they do not always have rational solutions. For details on 

 the case n = 3, m = 2, and Lagrange's extension of the method in his addition 

 IX to Euler's Algebra where a n = 1 is replaced by any equation of degree n, 

 see papers 161-6 of Ch. XXI; also Ch. XX. 



Lagrange 143 treated the problem to make y = p/q an integer when 

 p = a+bx-\ ---- , q = a l -\-b l x-i ---- are polynomials in x. By eliminating x, 



137 Unterrichtsblatter Math., 16, 1910, 41, 41-2. 

 Ibid., 17, 1911, 14-15. 



139 L'interm<diaire des math., 18, 1911, 214-6. 



140 Mathesis, (4), 3, 1913, 119-120. 



141 Math. Quest. Educat. Times, (2), 25, 1914, 61-3. 



142 M6m. Acad. R. Sc. Berlin, 23, ann^e 1767, 1769; Oeuvres, II, 527-532. Exposition by 



A. Desboves, Nouv. Ann. Math., (2), 18, 1879, 265-79; applications, 398-410, 433-444, 

 481-499; also by R. D. Carmichael, Diophantine Analysis, New York, 1915, 35-63. 

 Cf . Dirichlet 1 '; also Libri*"- 5 of Ch. XXV. 



143 Addition IV to Euler's Algebra, 2, 1774, 527-533. Oeuvres de Lagrange, VII, 95-8. 



Euler's Opera Omnia, (1), I, 579. 



