Going backwards with this highlights how to make lots of interesting problems

The golden ratio z is the largest solution to
z^2 = z + 1.
If we think of this number as the ratio a/b of two positive numbers a and b, then this equation can be rewritten as:
(a/b)^2 = (a/b) + 1
a^2 = ab + b^2 (*)

So any positive solution pair (a,b) to this will hand us the golden ratio. Now we can name
x = b^2, y = ab, and x + y = a^2.
Note that this system of equations is equivalent to (*) and it has the twist that y/x = a/b = z, the golden ratio.

Now think about the choices we can make for a and b to get an equivalent system of
f(x) = g(y) = h(x+y).
The choice Silver makes is a = 3^t and b = 4^t; solving for t gives:
log(3^2)(x) = log(3*4)(y) = log(4^2)(x + y)

Another interesting choice is to let a = z^2 and b = z, giving
x^(1/2) = y^(1/3) = (x + y)^(1/4)

Edited on December 9, 2004, 5:08 pm

Posted by owl on 2004-12-09 17:07:29