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.