Each of A and B is a positive real number and N is an integer with N > 1 satisfying:
AN - A - 1 = 0, and:
B2N - 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.