# The slope of $nx\ \%\ m$

(There is a follow-up question at MO.)

Let $$x\ \%\ m$$ be the residue of $$x$$ modulo $$m$$, i.e.

$$x \equiv x\ \%\ m\pmod{m}$$

Let $$\mu^n_m(x)$$ denote multiplication by $$n$$ modulo $$m$$, i.e.

$$\mu^n_m(x) = nx\ \%\ m$$

Plotting $$\mu^n_m(x)$$ for $$0 < n < m$$ yields patterns with characteristic "slopes", here for $$m=64$$:

[click image to enlarge]

While one could define the slope of $$\mu^n_m(x)$$ to be just $$n$$, this is not, what the patterns reveal. So I was looking for a more appropriate definition of the slope of $$\mu^n_m(x)$$ and came up with the following definition:

Let $$n_0$$ be the number greater than $$1$$ which minimizes $$(n_0 - 1)^2 + (\mu^n_m(n_0) - n)^2$$. Define then the slope $$s^n_m$$ of $$\mu^n_m(x)$$ to be the number

$$s^n_m = (\mu^n_m(n_0) - n)/(n_0 - 1)\pmod{m},$$

the latter being a shortcut for

$$((n_0 - 1)s^n_m - \mu^n_m(n_0) + n)\ \%\ m = 0$$

I've chosen this definition for two reasons:

1. Part 1 (the definition of $$n_0$$) considers the fact, that the dominant lines in the plot of $$\mu^n_m(x)$$ (which "show" the slope of the pattern) are those with a minimal number of parallels, i.e. a maximal distance between parallels, and thus the highest number of points on them, i.e. a minimal distance of points on them.

2. You can apply the same definition for the non-modular case: the number $$n_0>1$$ that minimizes $$(n_0 - 1)^2 + (n\cdot n_0 - n)^2 = (n_0 - 1)^2(1 + n^2)$$ is always $$2$$ – independently of $$n$$ –, and the slope is accordingly $$(n\cdot n_0 - n)/(n_0 - 1) = n$$.

Find here the values for $$n_0$$ and $$s^n_m$$ for $$m = 64$$ and $$0 < n < m$$ together with the "visual" slopes of the dominant lines:

My question is:

Are there closed expressions for the values of $$n_0$$ and $$s^n_m$$ as defined above – as number theoretic functions of $$n$$ and $$m$$?

Furthermore:

What can be generally said about the slope $$s^n_m$$? When is it $$0$$ or $$1$$? When does it "jump" when going from $$n$$ to $$n+1$$?

To make clear that no regular pattern and no simple closed expression for $$s^n_m$$ is to be expected (except for $$m$$ prime), I plotted $$s^n_{256}$$ as shades of gray, from white for $$s^n_{256} = 0$$ to black for $$s^{255}_{256} = 255$$:

Observations for $$m=64$$:

$$s^n_m = n$$ with these exceptions

• $$s^n_m = 0$$ for $$n = k\cdot m/8$$ with $$k = 1,\dots,7$$

• $$s^n_m = 1$$ for $$n = k\cdot m/4 + 1$$ with $$k = 0,\dots,3$$

• $$s^n_m = n - m/2$$ for $$n = 34, 35, 47, 53,54$$

Not very surprisingly one finds for prime numbers $$p$$ that $$s^n_p = n$$ for all $$n$$, e.g. for $$p=61$$:

Stated otherwise: $$s^n_m = n$$ for all $$n < m$$ iff $$\mathbb{Z}/m\mathbb{Z}$$ is a field.

In this respect, prime numbers behave like integers in non-modular arithemtic:

$$s^{n_1}_p = s^{n_2}_p \equiv n_1 = n_2$$

What's missing is a proof!