## Diophantine equation : (a-b)/(a+b) + (b-c)/(b+c) + (c-d)/(c+d) + (d-a)/(d+a) = 0

Determine integral solutions of the Diophantine equation

where   $a, \; b, \; c, \; d$   are distinct

Solutions:

$b \; \neq \; 0$,     $d \; = \; (a \, c)/b$,     $a^2 \, b \, c+a^2 \, c^2+a \,b^2 \, c+a \, b \, c^2 \; \neq \; 0$

If   $b = 1$,   then     $d \; = \; a \, c$,     $a^2 \, c + a^2 \, c^2 + a \, c + a \,c^2 \; \neq \; 0$

If   $b = 2$,   then     $d = (a \, c)/2$,     $a^2 \, c^2 + 2 \, a^2 \, c + 2 \, a \, c^2 + 4 \, a \, c \; \neq \; 0$

math grad - Interest: Number theory
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### 3 Responses to Diophantine equation : (a-b)/(a+b) + (b-c)/(b+c) + (c-d)/(c+d) + (d-a)/(d+a) = 0

1. Paul says:

Here are a few

Format {a, b, c, d}

{1280,16,75,6000}
{-100439,1,2,-200878}
{-101915,1,48,-4891920}
{-101023,1,59,-5960357}
{1532,1,-98,-150136}
{668,1,44,29392}
{528,1,79,41712}
{-100000,32,-31,96875}
{208,-112,-35,65}
{-100003,1,86,-8600258}
{-100000,1,39,-3900000}
{-100118,1,22,-2202596}
{1508,14,-112,-12064}
{-99000,6,5,-82500}
{-100455,1,30,-3013650}
{-100455,30,122,-408517}
{-100968,1,8,-807744}
{-98344,12,9,-73758}
{1490,10,96,14304}
{1768,13,-4,-544}

Paul.

• benvitalis says:

$d = (a \, c) \,/ \,b$
where   $a, b, c, d$   are distinct integers

• benvitalis says:

I posted above general solutions for b = 1 and b = 2