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oai:opus.uleth.ca:10133/48362017-04-28T20:00:44Zcom_10133_3346com_10133_1com_10133_296col_10133_3347col_10133_298

Akbary-Majdabadno, Amir Siavashi, Sahar University of Lethbridge. Faculty of Arts and Science 2017-04-28T18:57:31Z 2017-04-28T18:57:31Z 2017 https://hdl.handle.net/10133/4836 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. en_US Lethbridge, Alta : University of Lethbridge, Dept. of Mathematics and Computer Science Arts and Science Department of Mathematics and Computer Science Thesis (University of Lethbridge. Faculty of Arts and Science) class number one congruences quadratic fields Wieferich numbers Wieferich primes On the solutions of certain congruences Thesis Masters 0405 Yes No open.access