History of the Theory op Numbers. 
[Chap. XXIII 
For integral solutions Xi there exist relatively prime integers et» satisfying 
—«ia;2+aiXi+i = 0 (i=2, • • -, n—1), a x {x x -x 2 )+a 2 x x -\ \-a n - x x x = 0, 
and conversely. Hence 
x x = ka x • • •«„_!, Xj = kai - • -a n - X {a x -\ ha n -i)/«y-i, 
k being chosen to make the æ’s integers. 
M. Lagoutinsky 131 treated (1) for the case in which a, a x , • • • have no 
common divisor. Call their l.c.m. A, and set A/a = k, A/ai — ki. Thus 
k—'Lki. Hence we take k x , • • •, k n to be any integers without a common 
divisor and find the l.c.m. A of these k/s and k = 'Zk i . Then the solution 
is a=A/k, a,i=A/ki. 
Z lige 132 solved axy+bx+cy+d=0 by multiplying by a. Thus ax J rc = P, 
ay+b — Q, where bc—ad = 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 André 122 follows. 
P. Whitworth 133 noted that each divisor of N 2 = (x—N){y—N) yields a 
solution of l/x+l/y = l/N. 
P. Zühlke 134 gave, for l/x-\-lly = 2/m, 2x—m — p, 2y—m = q, pq = m 2 . If 
m is odd the resulting x, y are integers. 
E. S6s 135 noted that the general solution of Ifx = l/xi+l/x 2 is 
x = ky x y 2 , x x = ky 1 {y 1 -\ r y 2 ) ) x 2 = ky 2 {y 1 +y 2 ), 
where y x , y 2 are any relatively prime integers. Calling such a solution 
irreducible if k = l, and setting x = p x 1 - • -p“", where p x , ■ • - , p v are distinct 
primes, we find that there are 2 1,-1 essentially distinct irreducible solutions 
belonging to a given x, with x 2 , x x counted the same as x X} x 2 ; in all, 
2 { Ô (l+2a*) + l| 
essentially distinct solutions belonging to x. For the complete solution of 
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 z x —ZZ x , z 2 =ZZ 2 , where 
Zi, Z 2 are relatively prime. Then 
*-A /= aZlZî