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Subject: "The amazing book of squares"     Previous Topic | Next Topic
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Conferences The CTK Exchange College math Topic #643
Reading Topic #643
Akash
guest
Aug-24-07, 07:13 AM (EST)
 
"The amazing book of squares"
 
   The following is a puzzle from the book of squares authored by fibonacci somewhere in 1200's...Though it has gone old, it'still has its power.

As a specia case for demonstration sake, consider (12^2+10^2).(3^2+4^2).. The expression equals u^2+v^2 for some natural numbers u and v..well, u got it..Find them.

More generally, u can always have (a^2+b^2).(c^2+d^2) = (u^2+v^2)..as an identity in integers in throughout.


One easy, though genius, proof which i am aware of uses complex numbers. Try doing it without complex numbers anywhere in the picture??
And then please let me know of ur solution..Its eagerly awaited

Finally, for the record Fibonacci already solved this problem without complex numbers. Sadly, i am not aware of hi solution though i have come across references which emphasize him solving this problem.


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alexbadmin
Charter Member
2074 posts
Aug-24-07, 03:45 PM (EST)
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1. "RE: The amazing book of squares"
In response to message #0
 
   Just write down the identity you refer to and prove it by simple verification.

(a2 + b2)(c2 + d2) = (ac + bd)2 + (ad - bc)2.

So the question is not how to prove it but how to arrive at it. ... Probably by looking.


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