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

224 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1985.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.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD