4. Solutions of x/y+y/z+z/x=n

Let's try to find the solutions of

x/y+y/z+z/x=n

in the range -100 ≤ n ≤ 100.

This equation is not symmetric for x, y, z, so we cannot assume,

0 < |x| ≤ |y| ≤ |z|.

So we have to analyze six permutations of x,y,z by four sign,

( x, y, z)
(-x, y, z)
( x,-y, z)
( x, y,-z)

total 24 cases, at a first glance.

... But we can reduce the cost of computation.
The equation x/y+y/z+z/x is not symmetric, but it is cyclic, so

(x,y,z), (y,z,x), (z,x,y) are equivalent
(z,y,x), (x,z,y), (y,x,z) are equivalent

so we should search only two cases, (x,y,z) and (z,y,x).

Program

 10   ' xyyzzx.ub : x/y+y/z+z/x=n
 20   for Z=1 to 1000
 30     for Y=1 to Z
 40       for X=1 to Y
 50         if and{X=Y,Y=Z} then 70 else R=fnSub(X,Y,Z)
 60         if and{X=Y,Y=Z} then 70 else R=fnSub(X,Z,Y)
 70         if or{X=Y,X=Z} then 90 else R=fnSub(-X,Y,Z)
 80         if or{X=Y,X=Z} then 90 else R=fnSub(-X,Z,Y)
 90         if or{X=Y,Y=Z} then 110 else R=fnSub(X,-Y,Z)
100         if or{X=Z,Y=Z} then 110 else R=fnSub(X,-Z,Y)
110         if or{X=Z,Y=Z} then 130 else R=fnSub(X,Y,-Z)
120         if or{X=Y,Y=Z} then 130 else R=fnSub(X,Z,-Y)
130       next X
140     next Y
150   next Z
160   end
170   fnSub(X,Y,Z)
180   N=X//Y+Y//Z+Z//X:if den(N)>1 then return(0)
190   if N=0 then return(0)
200   if abs(N)>1000 then return(0)
210   if gcd(X,Y,Z)>1 then return(0)
220   print N;":";X;",";Y;",";Z
230   return(0)

Results of computation

Solutions for -100 ≤ n ≤ -1
Solutions for 1 ≤ n ≤ 100

We can get the solutions of x/y+y/z+z/x=n using elliptic curve.
Transform this equation as u/v+v+1/u=n. Under the following birational map,

X = -u
Y = uv

this equation is equivalent to the following elliptic curve E,

E : Y²+nXY=X³

Opposite direction is,

u = -X
v = -Y/X

Results of computation

Solutions for -100 ≤ n ≤ -1
Solutions for 1 ≤ n ≤ 100

Now we have got;

n = -48 : (x,y,z) = (72072752816411426700, 33132848506525529596688, -2507202774146263930905)
n = 62 : (x,y,z) = (4467832378776170000, -51609086900999886977, 278221158496143039700)