Sums of exceptional units in residue class rings

Abstract

Given a commutative ring R with 1 ∈ R and the multiplicative group R * of units, an element u ∈ R * is called an exceptional unit if 1 − u ∈ R * , i.e., if there is a u ∈ R * such that u + u = 1. We study the case R = Z n := Z/nZ of residue classes mod n and determine the number of representations of an arbitrary element c ∈ Z n as the sum of two exceptional units. As a consequence, we obtain the sumset Z * * n +Z * * n for all positive integers n, with Z * * n denoting the set of exceptional units of Z n .

On the sumsets of exceptional units in $$\mathbb {Z}_n$$ Z n

“…Recently, Sander [14] gave an exact formula for ϕ 2 (n, c). In order to present Sander's result, we define a function ϕ * * (n, c) below.…”

Sums of Polynomial-Type Exceptional Units Modulo

“…In [2], Anand et al presented a formula for N 2, f ,c (n) which also extends Sander's theorem [10]. However, it still remains open to give an explicit formula for…”

“…Exceptional units also have connections with cyclic resultants [14,15] and Lehmer's conjecture related to Mahler measure [11,12]. Following Sander's notation in [10], we use the coinage exunit to stand for exceptional unit. As usual, for any integer m and prime number p, we let ν p (m) stand for the p-adic valuation of m, that is, ν p (m) is the unique nonnegative integer r such that p r | m and p r+1 m. We denote by ω(m) := p prime,p|m 1 the number of distinct prime divisors of m. Yang and Zhao [18] extended Sander's result [10] by showing that the number of ways to represent each element of Z n as the sum of k exceptional units is given by…”

On the sums of squares of exceptional units in residue class rings

“…Obviously, N k,c,e (1) = 1. In 2016, Sander [12] obtained a formula for N 2,c,1 (n). One year later, Yang and Zhao [21] generalized Sander's result by giving the following explicit formula:…”