## Chapter 3 : n=(x+y+z)(1/x+1/y+1/z)

(version 1 : April 30, 1997; version 2 : April 09, 2008) [Japanese]

### Abstract

for given n, the Diophantine equation n=(x+y+z)(1/x+1/y+1/z) has integer solutions or not.

(Andrew Bremner , Richard K. Guy , and Richard J. Nowakowski ,
"Which integers are representable as the product of the sum of three integers with the sum of their reciprocals ?"
Math. Comp. Vol.61, Number 203, (1993 July)

In this chapter, first start from brute force method, try to find the solutions of -100 ≤ n ≤ 100 and understand the difficulty of this problems.
Next, refer to the above paper, find the solution by searching the rational point on birational elliptic curve.
Using this method, we can find the large solutions;

n = -100 : (x,y,z) = (4450012553, 219887106322, -663397965750)
n = 94 : (x,y,z) = (571064, -1799160, 79045681)

And also try to find the solutions of Diophantine equation x/y+y/z+z/x=n.
With elliptic curve, we can find the large solutions;

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

In order to understand the difficulty of this problem, please try to find the answers of the following equations.

• Q1 : -12=(x+y+z)(1/x+1/y+1/z)
• Q2 : 37=(x+y+z)(1/x+1/y+1/z)
• Q3 : -17=x/y+y/z+z/x
• Q4 : 14=x/y+y/z+z/x

### Contents

E-mail : kc2h-msm@asahi-net.or.jp
Hisanori Mishima