Distance between consecutive elements of the multiplicative group of integers modulo n

Abstract

For a prime number $p$, we consider its primorial $P:=p\#$ and $U(P):={\left(\ZZ{P}\right)}^\times$ the set of elements of the multiplicative group of integers modulo $P$ which we represent as points anticlockwise on a circle of perimeter $P$. These points considered with wrap around modulo $P$ are those not marked by the Eratosthenes sieve algorithm applied to all primes less than or equal to $p$. In this paper, we are mostly concerned with providing formulas to count the number of gaps of a given even length $D$ in $U(P)$ which we note $K(D,P)$. This work, presented with different notations is closely related to [5]. We prove the formulas in three steps. Although only the last step relates to the problem of gaps in the Eratosthenes sieve (see Section 3.2.2) the previous formulas may be of interest to study occurrences of defined gaps sequences. <ul> <li>For a positive integer $n$, we prove a general formula based on the inclusion-exclusion principle to count the number of occurrences of configurations<sup>1</sup> in any subset of $\ZZ{n}$. (see Equation (7) in Theorem 2.1).</li> <li>For a square-free integer $P$, we particularize this formula when the subset of interest is $U(P)$. (see Equation (11) in Theorem 3.2).</li> <li>For a prime $p$ and its primorial $P:=p\#$, we particularize the formula again to study gaps in $U(P)$. Given a positive integer $D$ representing a distance on the circle, we give formulas to count $K(D,P)$ the number of gaps of length $D$ between elements of $U(P)$ (see Equation (15) and Section 4.1).</li> </ul> In addition, we provide a formula (see Equation (27) in Theorem 5.1) to count the number of occurrences of gaps of an even length $N$ that contain exactly $i$ elements of $U(P)$. <sup>1</sup> <small>A defined sequence of gaps between the elements of the subset; this is referred to as a constellation in [5].</small>