(ab)^2 + (bc)^2 + (ca)^2 = d^2 and gcd(a,b,c) = 1

 

 
Find an infinite family of solutions to

(a \, b)^2 \; + \; (b \, c)^2 \; + \; (c \, a)^2 \; = \; d^2

such that   a, \; b, \; c,   and   d   are integers and   gcd(a,b,c) = 1

 
 

If   a = 1,   b = 2,    then   5 \, c^2 \; + \; 4 \; = \; d^2

 
(1)

FIBO EQ 1

 
(2)

FIBO EQ 2

 
(3)

FIBO EQ 3

 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

2 Responses to (ab)^2 + (bc)^2 + (ca)^2 = d^2 and gcd(a,b,c) = 1

  1. Paul says:

    Here is one of many, Format {a, b, c, n^2w}, here c is from the recurrence {3, -1}

    {1,2,3,49}
    {1,2,8,324}
    {1,2,21,2209}
    {1,2,55,15129}
    {1,2,144,103684}
    {1,2,377,710649}
    {1,2,987,4870849}
    {1,2,2584,33385284}
    {1,2,6765,228826129}
    {1,2,17711,1568397609}

    Paul.

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