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Title:  The Divisibility and Modular Properties of KthOrder Linear Recurrences Over The Ring of Integers of an Algebraic Number Field With Respect to Prime Ideals 
Author(s):  Somer, Lawrence Eric 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  Let K be an algebraic number field and R its ring of integers. Let k (GREATERTHEQ) 2 and let (w) be a kthorder linear recurrence over R satisfying the recursion relation (1) w(,n+k) = a(,1)w(,n+k1) + a(,2)w(,n+k2) +...+ a(,k)w(,n). Those recurrences (u) satisfying (1) for which u(,0) = u(,1) =...= u(,k2) = 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 kthorder 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 dpseudoprimes, where d is a positive integer such that 4 (VBAR) d. A Fermat dpseudoprime is an odd integer N for which there exists an integer a whose exponent modulo N is (N  1)/d. A Lucas dpseudoprime is an odd integer N for which there exists a secondorder unit sequence for which the rank of apparition of N is (n  (D/N))/d. All composite dpseudoprimes are determined when d = 2, 3, 5, or 6. All composite Fermat 7pseudoprimes are also found. 
Issue Date:  1985 
Type:  Text 
Description:  224 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1985. 
URI:  http://hdl.handle.net/2142/71235 
Other Identifier(s):  (UMI)AAI8521883 
Date Available in IDEALS:  20141216 
Date Deposited:  1985 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois