Counting sums of exceptional units in $\mathbb{Z}_n$

1.   Introduction

Let $R$ be a commutative ring with the identity $1_{R}$, and let $R^*$ be the multiplicative group consisting of all the units in $R$. An element $a\in R^*$ is said to be an exceptional unit if $1_R-a\in R^*$, i.e., if $a-1_R\in R^*$, or, in other words, if there exists a $b\in R^*$ satisfying $a+b = 1_R$. In 1969, exceptional units were first introduced by Nagell ^[7] to study certain cubic diophantine equations. From then on, many types of diophantine equations have been studied by means of exceptional units, for example, Thue equations ^[15], Thue-Mahler equations ^[16], and discriminant form equations ^[12].

Exceptional units also became a useful tool in number theory. For example, in 1977, Lenstra ^[5] introduced a new method to find Euclidean number fields by using exceptional units. Later on, many new Euclidean number fields were found with this method (see ^[4,6]). Furthermore, exceptional units also have connections with cyclic resultants ^[13,14] and Lehmer's conjecture related to Mahler's measure^[10,11].

Let $\mathbb{Z}, \ \mathbb{Z}^+$, and $\mathbb{P}$ be the sets of integers, positive integers, and primes, respectively. For $n\in\mathbb{Z}^+$, let $\mathbb{Z}_n = \{0, 1, \dots, n-1\}$ be the ring of residue classes modulo $n$. By definition, one has $\mathbb{Z}_n^* = \{a\in\mathbb{Z}_n: \gcd(a, n) = 1\}$. In this note, we set $\mathbb{Z}_n^{**}$ to be the set of all exceptional units in $\mathbb{Z}_n$, i.e., $\mathbb{Z}_n^{**}: = \{a\in\mathbb{Z}_n: \gcd(a, n) = \gcd(a-1, n) = 1\}$. Given $p\in\mathbb{P}$, we denote by $\nu_{p}(n)$ the $p$-adic valuation of $n$, i.e., $\nu_p(n)$ is the unique nonnegative integer $r$ satisfying $p^r|n$ and $p^{r+1}\nmid n$. Moreover, we let $\xi_n$ stand for the primitive $n$-th root of unity, i.e., $\xi_n: = e^{2\pi i/n}$.

In 2010, Harrington and Jones ^[3] obtained the following identity:

This result can also be deduced immediately from the theorems of Deaconescu ^[2] or Sander ^[8]. By the definition of an exceptional unit, we can see that

For $c\in\mathbb{Z}_n$, in 2009, it was proved by Sander ^[8] that

In this paper, we shall describe the elements in $\mathbb{Z}_n$, which could be written as the sum of one or more expected units. In addition, for these elements, we will derive the number of representations as such a sum. More specifically, for $n, s\in \mathbb{Z}^+$, and $c\in\mathbb{Z}_n$, we set

In 2016, Sander ^[9] presented an explicit formula for ${\mathcal N}_{2}(n, c)$. Now, we state Sander's theorem as follows:

Theorem 1.1. (Sander ^[8]) Given $n, k\in\mathbb{Z}^+$ and $c\in\mathbb{Z}_n$. The number ${\mathcal N}_{2}(n, c)$ satisfies the following relations:

while for all primes $p\geqslant 5$,

Let $\omega(n): = \sum_{p|n, p\in\mathbb{P}}1$ be the number of distinct prime divisors of $n$. In 2017, Yang and Zhao ^[17] extended Sander's theorem to finite terms by means of exponential sums, as below.

Theorem 1.2. (Yang and Zhao ^[17]) For $n, s\in\mathbb{Z}_{\geqslant 2}^+$ and $c\in\mathbb{Z}_n$, we have

In this paper, by using matrix theory, we give the following two results:

Theorem 1.3. Let $p\in\mathbb{P}, s\in\mathbb{Z}^+$, and let $\xi_{j}: = e^{2\pi i/j}$. Then

The second main result of this paper is the following corollary:

Corollary 1.1. Let $n, s\in\mathbb{Z}_{\geqslant 2}^+$ and $c\in\mathbb{Z}_n$. We have

where $\mathcal{N}_s(p, c)$ is determined by Theorem 1.3.

This paper is organized as follows: Section 2 provides several lemmas that are needed in the proof of Theorem 1.3 and Corollary 1.1. Then we give the proofs of Theorem 1.3 and Corollary 1.1 in Section 3.

2.   Preliminary lemmas

In this section, we supply several lemmas that will be needed in the proof of Theorem 1.3 and Corollary 1.1. We begin with the following result, which can be proved by using the Chinese remainder theorem:

Lemma 2.1. ^[1] Let $k, s\in\mathbb{Z}^+$, $f(x_1, ..., x_s)\in\mathbb{Z} [x_1, ..., x_s]$, and let $m_1, ..., m_k$ be pairwise relatively prime positive integers. For any integer $j$ with $1\le j\le k$, let $N_j$ be the number of zeros of $f(x_1, ..., x_s)\equiv 0\pmod {m_j}$, and let $N$ denote the number of zeros of $f(x_1, ..., x_s)\equiv 0 \pmod {\prod_{j = 1}^k m_j}$. Then $N = \prod_{j = 1}^k N_j$.

Lemma 2.2. Let $k\in\mathbb{Z}^+, p\in\mathbb{P}$. For any integer $c$, we have $\mathcal{N}_s(p^{k+1}, c) = p^{s-1}\mathcal{N}_s(p^k, c).$

Proof. Let $(b_1, \cdots, b_s)$ be a solution of $x_1+\cdots+x_s\equiv c\pmod {p^k}$, with $b_j\ (1\leq j\leq s)$ being exceptional units. One has $\gcd(b_j, p) = 1$. Let $b_1+\cdots+b_s-c = ap^k$ for some $a\in\mathbb{Z}$. For $k_1, \cdots, k_s\in\mathbb{Z}_{p^k}$, the congruence

holds if and only if

Clearly, the number of solutions to (2.1) is $p^{s-1}$.

Thus, one get $\mathcal{N}_s(p^{k+1}, c) = p^{s-1}\mathcal{N}_s(p^k, c)$. □

In this paper, we view vector $v$ as a column vector and $v^T$ as the transpose of $v$. For $a\in\mathbb{Z}$, we let $< a > _m$ denote the unique integer $r$ such that $r\equiv a\pmod m$ with $0\leqslant r\leqslant m-1$.

Definition 2.1. Let $v = (a_0, \cdots, a_{m-1})^T$ be a complex vector. The circulant matrix $A_v$ associated with $v$ is a $m\times m$ complex matrix having the form

In other words, if we let $A_v = (A_{i, j})$, then $A_{i, j} = a_{ < j-i > _m}$.

Lemma 2.3. Let $A_v$ be a circulant matrix associated to the vector $v = (a_0, \cdots, a_{m-1})^T$, and let $f(x) = \sum_{i = 0}^{m-1}{a_ix^i}$. Then, for each $j = 0, 1, \cdots, m-1$, $f(\xi_m^j)$ is an eigenvalue of $A_v$ and $v_j = (1, \xi_m^j, \xi_m^{2j}, \cdots, \xi_m^{j(m-1)})^T$ is an eigenvector corresponding to $f(\xi_m^j)$.

Proof. Let $\omega$ be any $m$-th root of unity. Set

Consider

Clearly,

For any $k\ge 2$, one has

Therefore, we obtain that

In particular, take $\omega = \xi_m^j$, where $j$ runs from $0$ to $m-1$. It then follows that $f(\xi_m^j)$ is an eigenvalue and

is an eigenvector corresponding to $f(\xi_m^j)$ for each $j = 0, 1, \cdots, m-1$.

This completes the proof of Lemma 2.3. □

Lemma 2.4. Let $k$ be a nonnegative integer and $m$ be a positive integer. Then

Proof. First, if $m\mid k$, then $\xi_m^{kj} = 1$ for any integer $j$. So

Next, we let $m\nmid k$. Then $k = qm+r$ for $0 < r < m$. Then one has

It follows that

The proof of Lemma 2.4 is complete. □

We also need the following result, which can be found in any standard linear algebra textbook.

Lemma 2.5. Let $A$ be a $m\times m$ matrix. Let $\lambda_1, \lambda_2, \cdots, \lambda_m$ be all the eigenvalues of $A$, and $\alpha_j$ be an eigenvector corresponding to $\lambda_j$ for every $1\leqslant j\leqslant m$. If $\alpha_1, \ \alpha_2, \cdots, \alpha_m$ are linearly independent, then $Q^{-1}AQ = \mathit{\text{diag}}(\lambda_1, \lambda_2, \cdots, \lambda_m)$ with $Q = (\alpha_1, \alpha_2, \cdots, \alpha_m)$.

Lemma 2.6. Let $V$ be a Vandermonde matrix of the form

Then $V$ is invertible, and

Proof. The proof follows from a direct calculation. □

3.   Proof of Theorem 1.3 and Corollary 1.1

Proof of Theorem 1.3. Let $(x_1, ..., x_s)\in (\mathbb{Z}_n^{**})^s$. It then follows that $x_s\ne 0$ and $x_s\ne 1$. Since

it is easy to see that for any integer $i$ with $0\le k\le p-1$, one has

That is,

It is clear that $A_v$ is a circulant matrix associated with the vector $v = (0, 1, \cdots, 1, 0)^T$. For simplicity, we set $\xi: = \xi_p$ in the following. Then $\xi_1: = \xi, \ \xi_2 = \xi^2, \cdots, \ \xi_{p-1} = \xi^{p-1}$ are all the primitive $p$-th roots of unity. Let $f(x) = x+x^2+\cdots+x^{p-2}$. By Lemma 2.3, for each $j = 0, 1, \cdots, p-1$, $f(\xi^j)$ is an eigenvalue of $A_v$ and $v_j = (1, \xi_1^j, \xi_2^{j}, \cdots, \xi_{p-1}^{j})^T$ is an eigenvector corresponding to the eigenvalue $f(\xi^j)$.

Let

Since $\det(B) = \prod\limits_{0\leqslant i < j\leqslant p-1}(\xi_j-\xi_i)\neq 0$, one has $v_0, v_1, \cdots, v_{p-1}$ are linearly independent. By Lemmas 2.5 and 2.6, we have

and

Notice that $\mathcal{N}_{1}(p, 0) = \mathcal{N}_{1}(p, 1) = 0$, $\mathcal{N}_{1}(p, j) = 1$ for $2\leqslant j\leqslant p-1$, and for $1\leqslant j\leqslant p-1$,

by Lemma 2.4. Therefore, one has

This completes the proof of Theorem 1.3.

Proof of Corollary 1.1. Let $n = \prod\limits_{p|n}p^{v_p(n)}$ be the canonical decomposition of $n$. By Lemmas 2.1 and 2.2, we get

This finishes the proof of Corollary 1.1.

4.   Conclusions

In the current study, by means of matrix theory, we present an explicit expression for $\sharp\Big\{(x_1, ..., x_s)\in (\mathbb{Z}_n^{**})^s : x_1^{k}+...+x_s^{k}\equiv c \pmod n\Big\}$ with $k = 1$. Naturally, one will ask for the formula for $\sharp\Big\{(x_1, ..., x_s)\in (\mathbb{Z}_n^{**})^s : x_1^{k}+...+x_s^{k}\equiv c \pmod n\Big\}$ with $k > 1$. Moreover, exceptional units are interesting and deserve further research.

Use of AI tools declaration

The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The author woulds like to thank the anonymous referees for their careful reading of the manuscript and helpful suggestions that improve the presentation of the paper.

Conflict of interest

The author declares that he have no conflict of interest.