On a question of f-exunits in $$\mathbb {Z}/n\mathbb {Z}$$

Autor: Bidisha Roy, Anand, Jaitra Chattopadhyay
Rok vydání: 2021
Předmět:
Zdroj: Archiv der Mathematik. 116:403-409
ISSN: 1420-8938
0003-889X
DOI: 10.1007/s00013-020-01558-w
Popis: In a commutative ring R with unity, a unit u is called exceptional if $$u-1$$ is also a unit. For $$R = {\mathbb {Z}}/n{\mathbb {Z}}$$ and for any $$f(X) \in {\mathbb {Z}}[X]$$ , an element $${\overline{u}} \in {\mathbb {Z}}/n{\mathbb {Z}}$$ is called an “f-exunit” if $$gcd(f(u),n) = 1$$ . Recently, we obtained the number of representations of a non-zero element of $${\mathbb {Z}}/n{\mathbb {Z}}$$ as a sum of two f-exunits for a particular infinite family of polynomials $$f(X) \in {\mathbb {Z}}[X]$$ . In this paper, we complete this problem by proving a similar formula for any non-constant polynomial $$f(X) \in {\mathbb {Z}}[X]$$ .
Databáze: OpenAIRE
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