Challenge Level

Show that there are no integer solutions $m, n$ of the equation $$\left({5\over 4}\right)^m = \left({2\over 1}\right)^n $$ which gives the number of major thirds in an octave on a musical scale.

Given integers $a, b, c $ and $d$ where $a$ and $b$ are coprime and $c$ and $d$ are coprime, find necessary and sufficient conditions for there to exist positive integers $m$ and $n$ such that $$\left({a\over b}\right)^m = \left({c\over d}\right)^n.$$

[The set of three problems Tuning and Ratio , Euclid's Algorithm and Musical Intervals and this problem Rarity were devised by Benjamin Wardaugh who used to be a member of the NRICH team. Benjamin is now doing research on the history of music and mathematics at Oxford University. Read Benjamin's article Music and Euclid's Algorithm for more on this subject.]