Solve in Integers

(5+3√2)^m = (3+5√2)ⁿ

The only solution is (m,n)=(0,0)

A^m = B^n
m*log(A) = n*log(B)
m/n = log(B)/log(A)

But log(5+3√2)/log(3+5√2) is clearly irrational so there are no integers m/n with n≠0.

A good approximation is 10117/9741 as supplied by
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfCALC.html#

Posted by Jer on 2013-05-14 12:45:34