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
 
diophantine-1

 
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
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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.

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