Each of A and B is a positive real number and N is an integer with N > 1 satisfying:

A^{N} - A - 1 = 0, and:

B^{2N} - B – 3A = 0

Which of A and B is greater?

Based on Charlie's work, I suspect that A > B for all N.

We know that A never equals B, because,

The first equation is A^N = A + 1

The second is B^2N = B + 3A.

Squaring the first and subtracting the 2nd gives

A^2n - B^2N = A^2 + -A - B + 1

If A = B they we have 0 = A^2 - 2A +1

This leads to A = 1, but this violates equation 1.