{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71235"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71235","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"The Divisibility and Modular Properties of Kth-Order Linear Recurrences Over The Ring of Integers of an Algebraic Number Field With Respect to Prime Ideals","abstract":"Let K be an algebraic number field and R its ring of integers. Let k (GREATERTHEQ) 2 and let (w) be a kth-order linear recurrence over R satisfying the recursion relation (1) w(,n+k) = a(,1)w(,n+k-1) + a(,2)w(,n+k-2) +...+ a(,k)w(,n). Those recurrences (u) satisfying (1) for which u(,0) = u(,1) =...= u(,k-2) = 0 and u(,k) = 1 are called unit sequences. Let f be the characteristic polynomial of the recurrence defined by (1). Let D be the discriminant of f. An ideal M is a maximal divisor of the kth-order recurrence (w) if the maximal number of successive terms of (w) it divides is k - 1.","abstract_html":"Let K be an algebraic number field and R its ring of integers. Let k (GREATERTHEQ) 2 and let (w) be a kth-order linear recurrence over R satisfying the recursion relation (1) w(,n+k) = a(,1)w(,n+k-1) + a(,2)w(,n+k-2) +...+ a(,k)w(,n). Those recurrences (u) satisfying (1) for which u(,0) = u(,1) =...= u(,k-2) = 0 and u(,k) = 1 are called unit sequences. Let f be the characteristic polynomial of the recurrence defined by (1). Let D be the discriminant of f. An ideal M is a maximal divisor of the kth-order recurrence (w) if the maximal number of successive terms of (w) it divides is k - 1.","abstract_has_math":false,"creators":["Somer, Lawrence Eric"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:13Z","date_published":"2014-12-16T06:18:13Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8521883"],"render_values":[{"text":"(UMI)AAI8521883","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71235","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Somer, Lawrence Eric"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:13Z","10000-01-01","1985"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71235","(UMI)AAI8521883"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Let K be an algebraic number field and R its ring of integers. Let k (GREATERTHEQ) 2 and let (w) be a kth-order linear recurrence over R satisfying the recursion relation (1) w(,n+k) = a(,1)w(,n+k-1) + a(,2)w(,n+k-2) +...+ a(,k)w(,n). Those recurrences (u) satisfying (1) for which u(,0) = u(,1) =...= u(,k-2) = 0 and u(,k) = 1 are called unit sequences. Let f be the characteristic polynomial of the recurrence defined by (1). Let D be the discriminant of f. An ideal M is a maximal divisor of the kth-order recurrence (w) if the maximal number of successive terms of (w) it divides is k - 1.","It is shown that, in general, the linear recurrence w(,n) (,n=0)('(INFIN)) has almost all prime ideals as maximal divisors if and only if the recur- rence has k - 1 consecutive terms equal to 0 when considered as the doubly infinite sequence w(,n) (,n=-(INFIN))('(INFIN)). Modular properties of kth- order unit sequences are considered with respect to prime ideals P. Constraints on (mu)(P), the period modulo P, and (beta)(P), the exponent of the multiplier modulo P, are determined for a unit sequence given (alpha)(P), the restricted period modulo P, and the exponent of a(,k) modulo P. Additional constraints are given for the possible values of (mu)(P), (alpha)(P), and (beta)(P) for a unit sequence in cases in which f either splits completely or remains irreducible modulo P. These additional constraints are also shown to be necessary and sufficient.","Improved primality tests are developed for an odd integer N for the case in which the factorization of N - 1 or N + 1 is completely known. These tests are based on the proof of the existence of only a finite number of composite Fermat and Lucas d-pseudoprimes, where d is a positive integer such that 4 (VBAR) d. A Fermat d-pseudoprime is an odd integer N for which there exists an integer a whose exponent modulo N is (N - 1)/d. A Lucas d-pseudoprime is an odd integer N for which there exists a second-order unit sequence for which the rank of apparition of N is (n - (D/N))/d. All composite d-pseudoprimes are determined when d = 2, 3, 5, or 6. All composite Fermat 7-pseudoprimes are also found.","Made available in DSpace on 2014-12-16T06:18:13Z (GMT). No. of bitstreams: 1 8521883.pdf: 5555423 bytes, checksum: 1bc223207cadbd1fbcabfe127bc50f16 (MD5) Previous issue date: 1985","Embargo set by: Seth Robbins for item 71401 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","224 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1985."]},{"key":"dc:title","label":"Title","values":["The Divisibility and Modular Properties of Kth-Order Linear Recurrences Over The Ring of Integers of an Algebraic Number Field With Respect to Prime Ideals"]}]}],"canonical_facts":{"dc:creator":["Somer, Lawrence Eric"],"dc:date":["2014-12-16T06:18:13Z","10000-01-01","1985"],"dc:description":["Let K be an algebraic number field and R its ring of integers. Let k (GREATERTHEQ) 2 and let (w) be a kth-order linear recurrence over R satisfying the recursion relation (1) w(,n+k) = a(,1)w(,n+k-1) + a(,2)w(,n+k-2) +...+ a(,k)w(,n). Those recurrences (u) satisfying (1) for which u(,0) = u(,1) =...= u(,k-2) = 0 and u(,k) = 1 are called unit sequences. Let f be the characteristic polynomial of the recurrence defined by (1). Let D be the discriminant of f. An ideal M is a maximal divisor of the kth-order recurrence (w) if the maximal number of successive terms of (w) it divides is k - 1.","It is shown that, in general, the linear recurrence w(,n) (,n=0)('(INFIN)) has almost all prime ideals as maximal divisors if and only if the recur- rence has k - 1 consecutive terms equal to 0 when considered as the doubly infinite sequence w(,n) (,n=-(INFIN))('(INFIN)). Modular properties of kth- order unit sequences are considered with respect to prime ideals P. Constraints on (mu)(P), the period modulo P, and (beta)(P), the exponent of the multiplier modulo P, are determined for a unit sequence given (alpha)(P), the restricted period modulo P, and the exponent of a(,k) modulo P. Additional constraints are given for the possible values of (mu)(P), (alpha)(P), and (beta)(P) for a unit sequence in cases in which f either splits completely or remains irreducible modulo P. These additional constraints are also shown to be necessary and sufficient.","Improved primality tests are developed for an odd integer N for the case in which the factorization of N - 1 or N + 1 is completely known. These tests are based on the proof of the existence of only a finite number of composite Fermat and Lucas d-pseudoprimes, where d is a positive integer such that 4 (VBAR) d. A Fermat d-pseudoprime is an odd integer N for which there exists an integer a whose exponent modulo N is (N - 1)/d. A Lucas d-pseudoprime is an odd integer N for which there exists a second-order unit sequence for which the rank of apparition of N is (n - (D/N))/d. All composite d-pseudoprimes are determined when d = 2, 3, 5, or 6. All composite Fermat 7-pseudoprimes are also found.","Made available in DSpace on 2014-12-16T06:18:13Z (GMT). No. of bitstreams: 1 8521883.pdf: 5555423 bytes, checksum: 1bc223207cadbd1fbcabfe127bc50f16 (MD5) Previous issue date: 1985","Embargo set by: Seth Robbins for item 71401 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","224 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1985."],"dc:identifier":["http://hdl.handle.net/2142/71235","(UMI)AAI8521883"],"dc:subject":["Mathematics"],"dc:title":["The Divisibility and Modular Properties of Kth-Order Linear Recurrences Over The Ring of Integers of an Algebraic Number Field With Respect to Prime Ideals"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:04Z"}