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



For integral solutions Xi there exist relatively prime integers a satisfying 

 a\Xz-t-a&i+i=Q (i = 2, - - , n 1), 

 and conversely. Hence 



k being chosen to make the x's integers. 



M. Lagoutinsky 131 treated (1) for the case in which a, 01, have no 

 common divisor. Call their l.c.m. A, and set Afa = k, A/ai = ki. Thus 

 k = 2/c,-. Hence we take ki, - - , k n to be any integers without a common 

 divisor and find the l.c.m. A of these fc/s and k = 2k{. Then the solution 

 is a = A/k, di=A{ki. 



Ziige 132 solved axy-}-bx-}-cy+d = by multiplying by a. Thus ax-\-c = P, 

 ay-{-b = Q, where bcad=PQ. For integral solutions, select the factors 

 P, Q so that P=c, Q^b (mod a). For the special case xy = a(x-\-y\ the 

 result by Andre* 122 follows. 



P. Whitworth 133 noted that each divisor of N 2 =(xN)(yN) yields a 

 solution of lfx+lfy = llN. 



P. Zuhlke 134 gave, for l/x+l/y = 2/m, 2xm = p, 2ym = q, pq = m?. If 

 m is odd the resulting x, y are integers. 



E. S6s 135 noted that the general solution of l/x = lfxi+l/x 2 is 



x = kyiy z , zi = 2/1(2/1+2/2), 2 = 2/2(2/1+2/2), 



where y\, y% are any relatively prime integers. Calling such a solution 

 irreducible if /c = l, and setting x = p* 1 - - -p a v v , where pi, , p v are distinct 

 primes, we find that there are 2"" 1 essentially distinct irreducible solutions 

 belonging to a given x, with x z , Xi counted the same as x\ t x z ; in all, 



essentially distinct solutions belonging to x. For the complete solution of 



rt l *+ 4- 1 



(2) - h "i > 



x x\ Xn 



2 n l parameters y f are introduced. 



S6s 136 noted that, if the a's are given integers, 



(3) .!+... +*J 



Z Zi Z n 



has (not the only) solutions z = ax, Zi = aiXi, if (2) holds. The complete 

 solution in positive integers, with g.c.d. unity, is obtained for (3). The 

 method is similar to that for the case n = 2. Set Zi = ZZi, z^ ZZ^ where 

 Z\ t Z 2 are relatively prime. Then 



131 L'interm6diaire des math., 7, 1900, 198. 

 13 * Archiv Math. Phys., (2), 17, 1900, 329-32. 



133 Math. Quest. Educ. Times, 75, 1901, 85. 



134 Archiv Math. Phys., (3), 8, 1905, 88. 



136 Zeitschrift Math. Naturw. Unterricht, 36, 1905, 97. 

 136 Ibid., 37, 1906, 186-190. 



