_{[I corrected the pictures and deleted one question due to user i707107’s valuable hint concerning cycles.]}

Visualizing the functions μn mod m(k)=kn mod m as graphs reveals lots of facts of modular arithmetic, among others the fixed points of μn mod m and the fact that μn mod m acts as a permutation πnm of [m]={0,1,…,m−1} iff n and m are coprime. Furthermore the cycle structure and spectrum of πnm can be visualized and related to number theoretic facts about n and m.

This is how the graph for μ3 mod 64(k)=3k mod 64 looks like when highlighting permutation cycles (the shorter the stronger):

When visualizing the function f2 mod m(k)=k2 mod m (which gives the quadratic residue of k modulo m) in the same way as a graph, other observations can be made and tried to relate to facts of number theory, esp. modular arithmetic:

These graphs are not as symmetric and regular than the graphs for μn mod m but observations can be made nevertheless:

the image of f2 mod m, i.e. those n with f2 mod m(k)=n for some k<m (black dots)

number and distribution of fixed points with f2 mod m(k)=k (fat black dots)

cycles with f2 mod m(n)(k)=k (colored lines)

parallel lines (not highlighted)

My questions are:

How can the symmetric distribution of image points n (with f2 mod 61(k)=n for some k, black dots in the picture below) be explained?

~~Can there be more than one cycle of length greater than 1 for f2 mod m?~~How does the length of the cycles depend on m?

How does the "parallel structure" depend on m?

With "parallel structure" I mean the number and size of groups of parallel lines. For example, f2 mod 8 has two groups of two parallel lines, f2 mod 12 has two groups of three parallel lines.

~~f2 mod 9 has no parallel lines.~~For f2 mod 61 one finds at least four groups of at least two parallel lines:

For other prime numbers m one finds no parallel lines at all, esp. for all primes m≤11 (for larger ones it is hard to tell).

**Answer**

This is not a complete answer to all of your questions. This is to show you some things you need to investigate. The first question is answered. The second question has an example. I do not know complete answers to the third and fourth questions, but I give a try on explaining your plot of m=61.

From your last sentences, it looks like you are interested in the case when m is a prime. Let m=p be an odd prime. Then consider p\equiv 1 mod 4, and p\equiv 3 mod 4.

In the former case p\equiv 1 mod 4, we see the symmetric black dots. This is because the Legendre symbol at -1 is 1. That is

\left( \frac{-1}p \right)=1.

This means -1 is a square of something in \mathbb{Z}/p\mathbb{Z}. Suppose x\equiv y^2 mod p, then we have -x \equiv z^2 mod p for some z\in\mathbb{Z}/p\mathbb{Z}.

Your example m=61 is a prime that is 1 mod 4. Thus, we have a symmetric black dots.

In general when p is an odd prime, the image of the square mapping is

\{ x^2 \ \mathrm{mod} \ p| 0\leq x \leq \frac{p-1}2 \}.

Note that the black dots represent image of the square mapping.

Thus, the number of black dots is \frac{p+1}2. In your example of m=61, we have 31 black dots.

Now we use a primitive root g in \mathbb{Z}/p\mathbb{Z}. Then any element x\in \mathbb{Z}/p\mathbb{Z} - \{0\}, we have some integer a such that x\equiv g^a mod p. Then a cycle formed by square mapping which includes x can be written as

\{g^{a\cdot 2^k} \ \mathrm{mod} \ p| k=0, 1, 2, \ldots \}.

To see if we have cycles, try solving

a\cdot 2^k \equiv a \ \mathrm{mod} \ p-1.

In your plot of m=61, we have a primitive root g=10 and the following are cycles of length greater than 1. All of these should be considered with modulo 61.

(g^{20} g^{40}),

(g^4 g^8 g^{16} g^{32}),

(g^{12} g^{24} g^{48} g^{36}),

(g^{56} g^{52} g^{44} g^{28})

I am not sure if you consider these as cycles, because there can be numbers in front of these, such as

g^5 \mapsto g^{10} \mapsto g^{20},

and comes in to the cycle (g^{20} g^{40}).

**Attribution***Source : Link , Question Author : Hans-Peter Stricker , Answer Author : Sungjin Kim*