A Criterion for Stability of Two-Term Recurrence Sequences Modulo 2k

Carlip, Walter; Jacobson, Eliot

doi:10.1006/ffta.1996.0023

Cited by 8 publications

(5 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(The coe cients which are not explicitly de ned, are assumed to bezero.) Furthermore, since Q 1 6 2 P n+1 x], there exists 0 i l, such that a i 6 2 P n+1 but a i 0 2 P n+1 for all 0 i 0 < i : Similarly, there exists 0 j m, such that b j 6 2 P k+1 but b j 0 2 P k+1 for all 0 j 0 < j : Then c i+j = a i+j b 0 + + a i+1 b j;1 + a i b j + a i;1 b j+1 + + a 0 b i+j : Since a 0 : : : a i;1 2 P n+1 and b 0 : : : b j;1 2 P k+1 thus a i+j b 0 : : : a i+1 b j;1 a i;1 b j+1 : : : a 0 b i+j 2 P n+k+1 : If c i+j 2 P n+k+1 , t h e n a i b j 2 P n+k+1 , but since P is a prime ideal, this contradicts the de nition of i and j, whereby c i+j 6 Lemma 3.6. Let F be a eld and u be a l.r.s.…”

Section: Results On Recurring Sequencesmentioning

confidence: 96%

See 1 more Smart Citation

Linear Recurring Sequences

Herendi¹

1994

Introduction to Finite Fields and Their Applications

View full text Add to dashboard Cite

Basic de nitions and resultsDedekind-domains are de ned in several ways in the literature. We will give t h e one which is the most suitable for our purposes.De nition 1.1. Let R be an integral domain. We call R a Dedekind-domain, if for every ideal I of R we can nd prime ideals P 1 : : : P k unique up to ordering, such that I = P 1 : : : P k .For general properties of Dedekind-domains see 7], 15], 16], 26], 35] and 48]. the companion matrix of u. Remark 1.15. With the above notations we haveu n (d) = M(u) n u 0 (d) which will be used frequently in the paper.Remark 1.16. We mention that if we reduce a linear recurring sequence modulo some ideal in a Dedekind-domain, then we get a linear recurring sequence in the residue class system, which may have di erent properties than the original sequence (e.g. the minimal order of the reduced sequence may became smaller). By Remark 1.16 it has sense to introduce the following notations:De nition 1.17. Let s be a positive integer. With the notation of De nition 1.14 d P (u s) will denote the minimal order, % P (u s) the minimal period length, M P (u s) the companion matrix and a s 0 : : : a s d P (u s);1 the de ning coecients corresponding to the minimal recurrence relation of u modulo P s . Remark 1.18. As far as there is no confusion, we will simplify our notation by omitting unnecessary parameters, for instance, by cancelling the sign of the ideal P.For further properties of linear recurring sequences we refer to 24].Chapter 2 Dedekind-domains and modulesFor the discussions of the later chapters we will need some special properties of Dedekind-domains. In this chapter we state all the results we will use. The material of this and the 4th chapter is a generalization of 19].Throughout the chapter let R be a Dedekind-domain, and let P be a prime ideal of R. Suppose that R=P has N(P) elements, and N(P) < 1. Since R is a Dedekind-domain, P is maximal and R=P is a ( nite) eld (see e.g. 16]). Hence, we know that N(P) = l with some rational prime and an integer l 1 (see e.g. 24]). First we turn to the determination of N(P k ). For this we n e e d t h e f o l l o wing: Theorem 2.1. Let k 2 N . Then the additive groups of the rings R=P and P k;1 =P k are isomorphic. Proof. See e.g. 26]. Corollary 2.2. Let k 2 N . Then N(P k ) = N(P) k .Proof. By the isomorphism theorem of groups, we know that the additive groups of (R=P k )=(P k;1 =P k ) and R=P k;1 are isomorphic, thus

show abstract

Section: Results On Recurring Sequencesmentioning

confidence: 96%

“…Suppose that u 0 : : : u d;1 is not a semi-basis modulo P s , whence by Theorem 2.14, they are strongly dependent. This yields that there exists a set of coe cients 0 : : : d;1 2 R and k 2 f 0 : : : d ; 1g, such that k 6 The smallest such a T will be denoted by T(u).…”

Section: Results On Recurring Sequencesmentioning

confidence: 99%

Linear Recurring Sequences

Herendi¹

1994

Introduction to Finite Fields and Their Applications

View full text Add to dashboard Cite

show abstract

“…In this paper we apply techniques similar to those used in [1] to characterize the stability of sequences whose parameter a is even.…”

Section: Introductionmentioning

confidence: 99%

“…In [1][2][3] we examined the stability modulo two of sequences for which the parameter a is odd and showed how stability leads to a precise description of the frequency distribution functions of such sequences. In this paper we apply techniques similar to those used in [1] to characterize the stability of sequences whose parameter a is even.…”

Section: Introductionmentioning

confidence: 99%

Stability of Two-Term Recurrence Sequences with Even Parameter

Carlip¹,

Jacobson

1997

Finite Fields and Their Applications

Self Cite

View full text Add to dashboard Cite

show abstract

“…In practice, as Carlip and Jacobson observed in [4], these computations may be arbitrarily long; the sets Ω(p r ) may be arbitrarily large and the constant N (the index of stability) required in the definition of stability also arbitrarily large. Stability of second-order recurrences modulo two has been extensively studied by Carlip and Jacobson in [2,3,4,5], while stability modulo odd primes has been examined by Carlip, Jacobson, and Somer in [6] and Carroll, Jacobson, and Somer in [9]. In recent work Carlip and Somer [7,21] have studied the frequency distributions of second-order recurrences modulo powers of odd primes.…”

Section: Introduction Let W(a B) = (W)mentioning

confidence: 99%