Linear recurrence sequences satisfying congruence conditions

Abstract

It is well-known that there exist integer linear recurrence sequences {x n } such that x p ≡ x 1 (mod p) for all primes p. It is less well-known, but still classical, that there exist such sequences satisfying the stronger condition x p n ≡ x p n−1 (mod p n ) for all primes p and n ≥ 1, or even m | d|m μ(m/d)x d for all m ≥ 1. These congruence conditions generalize Fermat's little theorem, Euler's theorem, and Gauss's congruence, respectively. In this paper we classify sequences of these three types. Our classification for the first type is in terms of linear dependencies of the characteristic zeros; for the second, it involves recurrence sequences vanishing on arithmetic progressions; and for the last type we give an explicit classification in terms of traces of powers.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 2338 G. MINTON Licensed to Univ of Nebraska-Lincoln. Prepared on Tue Feb 3 14:02:53 EST 2015 for download from IP 129.93.16.3.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use s}. Then the dimension of the vector space of Fermat sequences with characteristic polynomial f (t) is exactlyRemark 2.5. In addition to computing the dimension, our forthcoming argument implicitly gives an algorithm for computing the space of Fermat sequences in terms of the space of vanishing Q-linear combinations of {r i }.splitting field. Let e, be as in Theorem 2.4, and let u = max{0, d − s − 1}. Then the vector space of Fermat sequences with characteristic polynomial f (t) has dimension exactly (s − e + ) + u.Proof. This follows from the combination of Proposition 2.3 and Theorem 2.4.In certain cases we can simplify the classification, for instance, as follows. Corollary 2.7. Let f (t) ∈ Q[t] be an irreducible polynomial. Suppose that either (i) deg f (t) is prime or (ii) the Galois group of f acts doubly-transitively on its zeros. Then, letting θ be a zero of f , the only Fermat sequences with characteristic polynomial f (t) are multiples of {TrWhen the conditions of Corollary 2.7 do not hold, we can sometimes find many more Fermat sequences. We discuss some examples of this in §5.In §6 we move on to Euler sequences, obtaining the following results.Proposition 2.8. Let {x n } be a rational linear recurrence sequence with separable decomposition x n = y n +nz n +w n . Let z n = z n +nz n be the separable decomposition of {z n }. Then {x n } is an Euler sequence iff {y n } is an Euler sequence, {z n } is a Fermat sequence, z 1 = 0, and z 1 + w 1 = 0.Definition 2.9. A trace sequence is a sequence {x n } of the formwhere K is an algebraic number field, a 1 , . . . , a r ∈ Q, and θ 1 , . . . , θ r ∈ K.Definition 2.10. A vanishing sequence is a rational linear recurrence sequence {x n } such that, for some integer m ≥ 1, x n = 0 for all n relatively prime to m. Theorem 2.11. Let {y n } be a rational linear recurrence sequence with separable characteristic polynomial. Then {y n } is an Euler sequence iff it is the...

q-Congruences, with applications to supercongruences and the cyclic sieving phenomenon

“…Steinlein [Ste17] and Corollary 3.4). Other terms for a sequence satisfying the Gauss congruences include 'Gauss sequence' [Gil89,Min14], 'generalized Fermat sequence' [DHL03], and 'Dold sequence' [JM06,Ch. 3.1].…”

Generalized Dold Sequences on Partially-Ordered Sets

“…Proof. Assume now the integer vector valued sequences A n and X n are joined by the relation (27), i.e.…”

Dold sequences, periodic points, and dynamics

“…The sequences which we will call Dold sequences as a result of the work of Dold [27] have (for example) also been called sequences having divisibility in the thesis of Moss [92], prerealisable sequences by Arias de Reyna [2], relatively realisable sequences by Neumärker [94], Gauss sequences by Minton [89], and generalised Fermat sequences or Fermat sequences by Du, Huang, and Li [28,29]. Doubtless there are other names, reflecting the long history and multiple settings in which they appear.…”

“…Theorem 2.4 (Minton[89, Theorem 2.15 and Remark 2.16]). An integer linear recurrence sequence (a n ) is a rational multiple of a Dold sequence if and only if it is a trace sequence, meaning that there is an algebraic number field K, rationals b 1 , .…”