So , , , etc. It is easy to see that
. It should be just as easy to see that for any m.
(Technical point- you must have in order that this be true.)
Notes
You may use the following formulae that a^{0 }= 1 and a^{-n}= 1 / a^{n} for any a ℚ \ {0} and n .
We also need to define the "closeness to 1" of a positive rational number, as follows. For any q Q+, let q' = q if q 1, but let q' = 1/q if q < 1 (so that q' 1 in all cases). Then, given q, r Q+ with 1 q' < r', we say that q is closer to 1 than r is.
The Question
The pitch P of a musical note is a number (actually the frequency of a sound wave, in some given units). Pythagoras is said to have discovered that two notes played together sound harmonious if their pitches are related by the ratio 3/2. Pythagorean tuning aims to generate a musical scale (a nite set of musical notes) by starting with a note with some arbitrary pitch P_{0}, then intorducing a note with the pitch P_{1 = }3/2(P_{0}), and further notes with P_{K+1 = }3/2(P_{k}) where k=1, 2, ... , N-1) for some N . The objective is to choose N such that P_{N } P_{0} (under the equivalence relation that is x y if x = 2^{m}*y for some m ).
Using the fundamental Theorem of Arithmetic, prove that it is impossible for P_{N } P_{0}for any N with this system of tuning.