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?

It is relatively easy to show that both A and B decrease when N grows bigger.

Looking at both equations one can see that the above values are never less than 1, each of them approach 1 as a limit, and we want to see which is bigger for the same N.

Assume A=k*B

And denote 2N by t

WLOG B=1+b

Value of b getting smaller as t increases

Since B^t-B=3k*B

and B^t>1+bT

1+tb-1-b=b*(t-1)

And for any N > 2 t-1>3

**k>b*(t-1)/(3*b) = (t-1)/3 >1** ergo ** ****A>B**

**as suspected**

*Edited on ***August 29, 2016, 1:46 am**