On the solutions of certain congruences
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Date
2017
Authors
Siavashi, Sahar
University of Lethbridge. Faculty of Arts and Science
Journal Title
Journal ISSN
Volume Title
Publisher
Lethbridge, Alta : University of Lethbridge, Dept. of Mathematics and Computer Science
Abstract
We study the solutions of certain congruences in different rings. The congruences include
a^p-1 ≡ 1 (mod p^2);
for integer a > 1 and prime p with p does not divide by a, and
a^φ(m) ≡ 1 (mod m^2),
for integer m with (a;m) = 1; where j is Euler’s totient function. The solutions of these
congruences lead to Wieferich primes and Wieferich numbers. In another direction this
thesis explores the extensions of these concepts to other number fields such as quadratic
fields of class number one. We also study the solutions of the congruence
g^m - g^n ≡ 0 (mod f^m - f^n);
where m and n are two distinct natural numbers and f and g are two relatively prime polynomials
with coefficients in the field of complex numbers.
Description
Keywords
class number one , congruences , quadratic fields , Wieferich numbers , Wieferich primes