Monday, April 6, 2009

An irrational logarithm

A short problem proposed by myself for a recent student team competition, held in Bowling Green:

Show that the logarithm in base 4+sqrt(5) of 6+sqrt(5) is irrational.
The proof proceeds by contradiction. Assume the logarithm in base 4+sqrt(5) of 6+sqrt(5) equals p/q with p,q positive integers. Then

(4+sqrt(5))^p=(6+sqrt(5))^q (*)
By taking conjugates,

(4-sqrt(5))^p=(6-sqrt(5))^q (**)
Multiplying (*) and (**) term by term we get

11^p=31^q.
The fundamental theorem of arithmetic tells us this cannot happen.