Math CentralQuandaries & Queries


Question from Simon:

I wish to find all the answers for the following equation over the interval (0,1):

cos^2(pi * n^x) + cos^2(pi * n^(1-x)) - 2 = 0
where n is any integer > 0

some notes: a solution containing an integer constant must be avoided - solving this equation numerically is not practical
i need an algebraic solution, calculus solution, anything that will let me arrive at a solution quickly
also note that, for my use, it is sufficient to use the interval (0,0.5) AND in my use there will only be one solution over that interval.

As this is most definitely not a textbook problem, i do not expect an answer, however, i thank you for any help you may provide


I doubt very much that you will find what you are looking for in fact I very much expect that there is no solution in terms of elementary functions.

I did experiment with some graphing software and, at least for small values of n, 1≤ n ≤ 20, I saw that the behaviour depended on n. For n = 2, 3, 5, 7, 11, 13, 17 and 19 it looked like there was no solution in (0, 1). For n = 4 and 9 the only solution is (0, 1) seems to be x = 1/2. For n = 6, 8, 10, 14, 15 and 16 it looks like there is one solution in (0, 1/2) and for n = 12, 18 and 20 there seem to be 2 solutions in (0, 1/2).

You didn't say how you want to use this but one possibility is to approximate the function f(x) = cos2(π × nx) + cos2(π × n1-x) - 2 using a power series. I am no expert on this and I don't even have a suggestion of the type of series to try.

I'm sorry I'm not much help,

Well, RD showed me wrong in a very nice way. Harley


It's not as hard as you may have thought. Assuming we're looking only at real numbers here, cos2(alpha) cannot be outside [0,1] for any alpha.

(1) So what can you say about

cos2(π × nx) and cos2(π × n1-x)?

(2) Then what can you say about

cos(π × nx) and cos(π × n1-x) ?

(3) And what does this tell you about nx and n1-x?

(4) How are nx and n1-x related? (Hint: what is their product?)

(5) You should now be able to see why there are (eg) two solutions in (0, 0.5) for n=12, and to write them as ratios of logarithms.

By the way, are solutions in (0, 0.5) really sufficient? If n is a perfect square, do you really not want the solution x=0.5)?

Good Hunting!

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