# number theory – Find roots of a specific discontinuous trigonometric function

Given the following function $$f(x)$$ and an arbitrary positive non-prime integer $$N$$

$${ f(x) = sin(Npi/x) + cos(2Npi/x) + |sin(3Npi/x)| + |cos(5Npi/x)| + 1, N in mathbb{Z}, x in mathbb{R} }$$

I am looking for any root in the range $$2 <= x < N$$, I am pretty sure all $$f(x)$$ are greater than or equal to zero, so the roots are global minima. All the roots are also points of discontinuity and their $$x in mathbb{Z}$$.

## Question

Is there any, even numerical way, to go about finding these roots?

Here is a plot of $$f(x)$$ for $$N=517$$. It isn’t very good, its hard to choose a zoom level which will demonstrate all the features. But you can see I marked a zero at $$x=22$$, you will notice it looks like there is a zero at $$x approx 26.5$$ but it is actually $$f(x) approx 0.0011$$ if one were to zoom in.

There are many of these “close” to zero points, but none are actually zero, in the range I’m interested in ($$2<=x), unless 2 conditions I have found are met:

1. $$x in mathbb{Z}$$
2. $$GCD(x,N) > 1$$

My example $$N=517$$ the roots are $$x in {-2,22,94}$$, we don’t care about $$-2$$ and $$GCD(22,517)=11$$ and $$GCD(94,517) = 47$$.

I don’t think the GCD observation will help in finding the roots, but I mention it because I think its interesting. Posted on Categories Articles