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

Abstract

Let R be a commutative ring with 1 ∈ R and R * be the multiplicative group of its units. In 1969, Nagell introduced the concept of an exceptional unit, namely a unit u such that 1 − u is also a unit. Let Z n be the ring of residue classes modulo n. In this paper, given an integer k ≥ 2, we obtain an exact formula for the number of ways to represent each element of Z n as the sum of k exceptional units. This generalizes a recent result of J. W. Sander for the case k = 2.

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

“…Evidently, Theorem 1.1 generalizes the Yang-Zhao theorem [21] from the liner case to the quadratic case, and extends Mollahajiaghaei's theorem [7] from the unit solution case to the exceptional unit solution case.…”

“…where ω(n) := p prime, p|n 1 stands for the number of distinct prime divisors of n. We point out that an error in the formula in Theorem 1 of [21] was corrected by Zhao et al [22], where the sign factor (−1) k should read (−1) kω(n) .…”

“…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:…”

On the additive and multiplicative structures of the exceptional units in finite commutative rings

“…In this paper, we consider the additive and multiplicative structures of the exceptional units of a finite commutative ring. We show that as in [6,11], there is an exact formula for the number of ways to represent a unit as a product of k exceptional units. As an application and in combination with Miguel's result, we completely determine the additive and multiplicative structures of such exceptional units in our situation (see Theorems 1.7 and 1.10 below); and then as an application we find a necessary and sufficient condition under which R is generated by its exceptional units (see Corollaries 1.8 and 1.11 below).…”

Pichia pastoris as a Versatile Cell Factory for the Production of Industrial Enzymes and Chemicals: Current Status and Future Perspectives

“…As a result, much effort has been directed toward the optimization of the expression levels of extracellular heterologous proteins. This paper is not intended to provide a comprehensive review of genetic engineering and process control strategies to improve protein production in P. pastoris (for more detailed review, refer to Yang et al). Rather we summarized the most commonly seen bottlenecks in pursuing industrial‐scale production of heterologous proteins and presented possible solutions to overcome them (Table ).…”