The Maximal Difference of Different Powers of an Element Modulo

Abstract

In this paper, we investigate the maximal difference of integer powers of an element modulo . Let denote the integer with such that for any integer . Using the bounds for exponential sums, we obtain a lower bound of the function which gives .

1. Introduction

Let be an integer and be an integer with . We know that there exists a unique integer such that , where is called the inverse of modulo with . We denote by . The difference between an element and its inverse modulo has been studied by several mathematicians.

Zhang [1] is the first person to explicitly study the distribution between an integer and its inverse modulo , proving thatwhere is the Euler function and is the divisor function.

Zhang [2] gives some further results for the distribution of . For any , he defines thatwhere denotes the number of the elements in and shows that

Meanwhile, Zheng [3] obtains similar results in a more general context. More generalizations of [1, 2] are in recent papers [4–7].

Our object in this paper is to study the maximal difference of different powers of an element modulo . The investigation for the maximal difference of different powers of modulo is motivated by the problem of the maximal difference of , where and run through the set such that . Khan [8] definesproving that

Later, Khan and Shaparlinski [9] show that

Let be an integer and denote residues’ modulo . In this paper, we assume these residues to consist of the elements . We write to be the set of elements relatively prime to . Let be the integer with such that for any integer . Define

We prove the following theorems.

Theorem 1. Letbe an integer. Then, for arbitrary given positive integers, we have

In fact, we obtain a more general result which gives a lower bound for :

Theorem 2. Letbe an integer. Then, for arbitrary given positive integers, we havewhere .

By applying Theorem 2, one can obtain the following.

Corollary 1. Letbe an integer. Then, for arbitrary given positive integers, we have

2. Preliminaries

2.1. Notations

In this paper, is the integer part of . We denote the group of units of by ; thus, the cardinality of is given by the Euler function . Let be the counting function of the number of distinct primes. The number and the sum of the divisor function are defined byand the Riemann zeta function is written by

In what follows, we use the Landau symbol with the understanding that any implied constants are absolute. For given functions and , the notation is equivalent to the statement that the inequality holds with some constant .

We recall the identity which follows from the formula of the sum of a geometric progression:where .

2.2. Several Lemmas

We need several lemmas to prove our theorems.

Lemma 1. For any, with, there holds

Proof. (see Lemma 17.3 in [10]).

Lemma 2. Letbe a prime andbe an integer with. Then, for any integersand, there holds

Proof. This can be easily obtained from formula (0.5) in [11].

Lemma 3. Letbe an integer andbe a fixed positive integer. For any integerwith, we have the following estimation:

Proof. Let be the prime factor decomposition of . Note thatIn fact, if and pass through a reduced residue system modulo and , respectively, passes through a reduced system modulo . One can writeThis yields identity (17). By Lemma 2, we get Lemma 3.

Lemma 4. Letbe fixed integers. Then, the bound,holds.

Proof. It is easily obtained from Lemma 1 in [12].

To introduce the following lemma better, we define

Lemma 5. For any, we have

Proof. We begin with the case when . Apparently, this indicates that and . From Lemma 1 and 3, we deriveUsing a similar method, we have the estimation,We now consider the case . By Lemma 1 and 4, one obtainsThis completes the proof.

2.3. Proofs of the Theorems

Now, we complete the proofs of our main results.

Proof. of Theorem 2. We denote by the number of the solution of congruences:In the following, we show the range of in the case that congruences (25) are solvable for , and integers . Then, it can be easily deduced thatApplying the exponential sums identity (13), we haveProvided that , the right-hand side of equation (27) is . Then, we haveFrom Lemma 5, one can writeWe knowRemark that , so we can get the inequalityAfter a simple calculation, one can writeThe right-hand side of (32) has to be positive. It suffices to show thatMultiplying both side of inequality (33) by and calculating briefly, we haveHence,This finishes the proof of Theorem 2.

Proof. of Corollary 1. Applying Theorem 2, one can obtainwhere we use the boundsThe proof of Corollary 1 is complete.

Proof. of Theorem 1. Combining Corollary 1 with the bounds,See Theorem 328 in [10]; Theorem 1 follows immediately.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Basic Research Program for Nature Science of Shaanxi Province (2014JM1001 and 2015KJXX-27) and N.S.F.(11 971 381, 11 471 258, and 11 701 447) of P. R. China.