A Recurrence Generating Multiples of Primes: 10655

Bencze, Mihály; Saracino, Dan; Stenger, Allen; Amghibech, Saïd; Anglesio, Jean; Bauer, Rudolf; Siegel, Andrew F.; Bloom, D. M.; Body, G. L.; Callan, David; Chapman, Robin; Ford, Kevin; Gagola, Stephen M.; Gauthier, Nicolas; Grounds, W. V.; Hagedorn, Thomas R.; Koether, Robb T.; Komanda, Nasha; Krishnan, Ramaswamy; Lau, Kee-Wai; Lindsey, John H.; Mattics, Leon; Minh, Canh; Ozsoylev, H. N.; Yildirim, C. Y.; Poonacha, P. G.; Sanjeev, N. R.; Popescu, Calin; Robertson, John P.; Sieffert, H.-J.; Shallit, Jeffrey; Singer, Nicholas C.; Stadler, Albert; Terr, D. C.; Triff, T. V.; Trimble, Todd; Trojovský, Pavel; Group, Anchorage Math Solutions; Group, GCHQ Problems; Group, NSA Problems

doi:10.2307/2589334

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“…In a recent elementary note [16] we surveyed three proofs of the following fact: the Perrin sequence {P n }, defined by P 1 = 0, P 2 = 2, P 3 = 3, and P n = P n−2 +P n−3 for n ≥ 4 (OEIS A001608 [17]), satisfies p | P p for all primes p. This sequence and close relatives (e.g., OEIS A050443 and A001634) have repeatedly appeared in the literature: in problems [5][6][7]21]; as sources for pseudoprimality tests [1,13]; and even in a popular comic strip [2]! The Perrin sequence is an example of a trace sequence, i.e., a sequence {x n } of the form x n = Tr Q(θ)/Q (θ n ) for an algebraic integer θ.…”

Section: Introductionmentioning

confidence: 99%

Linear recurrence sequences satisfying congruence conditions

Minton

2014

Proc. Amer. Math. Soc.

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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...

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Section: Introductionmentioning

confidence: 99%

Linear recurrence sequences satisfying congruence conditions

Minton

2014

Proc. Amer. Math. Soc.

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A Recurrence Generating Multiples of Primes: 10655

Cited by 1 publication

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Linear recurrence sequences satisfying congruence conditions

Linear recurrence sequences satisfying congruence conditions

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