{"id":{"repo_id":"lethbridge","oai_identifier":"oai:opus.uleth.ca:10133/4836"},"canonical_url":"https://search.dev.ndltd.org/etd/lethbridge/oai:opus.uleth.ca:10133/4836","repository":{"repo_id":"lethbridge","name":"University of Lethbridge","base_url":"https://opus.uleth.ca/server/oai/request"},"display":{"title":"On the solutions of certain congruences","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Siavashi, Sahar","University of Lethbridge. 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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."]},{"key":"dc:description.other","label":"Dc Description Other","values":["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."]},{"key":"dc:title","label":"Title","values":["On the solutions of certain congruences"]}]}],"canonical_facts":{"dc:contributor.supervisor":["Akbary-Majdabadno, Amir"],"dc:creator":["Siavashi, Sahar","University of Lethbridge. Faculty of Arts and Science"],"dc:date.accessioned":["2017-04-28T18:57:31Z"],"dc:date.available":["2017-04-28T18:57:31Z"],"dc:date.issued":["2017"],"dc:description.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."],"dc:description.other":["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."],"dc:identifier":["hdl:10133/4836"],"dc:identifier.uri":["https://hdl.handle.net/10133/4836"],"dc:language.iso":["en_US"],"dc:publisher":["Lethbridge, Alta : University of Lethbridge, Dept. of Mathematics and Computer Science"],"dc:publisher.department":["Department of Mathematics and Computer Science"],"dc:subject":["class number one","congruences","quadratic fields","Wieferich numbers","Wieferich primes"],"dc:title":["On the solutions of certain congruences"],"dc:type":["Thesis"]},"updated_at":"2026-08-21T16:45:55Z"}