(In reply to
Isn't this more general and simple? by Jer)
Hah! I started with just that idea in mind, but then got distracted. You are correct; thanks for putting me back on the right track.
The generalisation I was originally aiming for also involves Fibonacci numbers F_(n), where a and b can be anything:
2*(F_(n1)a+F_(n)b)*(F_(n)a+F_(n+1)b) = x
(F_(n2)a+F_(n1)b)*(F_(n+1)a+F_(n+2)b) = y
(F_(n1)a+F_(n)b)^2+(F_(n)a+F_(n+1)b)^2 = z
then x^2+y^2=z^2.
Small values of c:
a^2+2ab+2b^2, 2a^2+6ab+5b^2, 5a^2+16ab+13b^2, 13a^2+42ab+34b^2, 34a^2+110ab+89b^2,..etc; generally:
F_(n1)a^2+2F_(n)ab+F_(n+1)b^2 .
When a=b=1, these expressions return the Fibonacci numbers F_(2n1) {5,13,34,89,...} themselves.
Very pretty.
Edited on June 27, 2014, 4:03 am

Posted by broll
on 20140627 02:09:59 