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ALGEBRAIC DIVISIBILITY SEQUENCES OVER FUNCTION FIELDS

Part of: Sequences and sets

Published online by Cambridge University Press:  24 April 2012

PATRICK INGRAM ,
VALÉRY MAHÉ ,
JOSEPH H. SILVERMAN ,
KATHERINE E. STANGE  and
MARCO STRENG
PATRICK INGRAM
Affiliation:
Department of Mathematics, Colorado State University, Fort Collins, CO 80521, USA (email: pingram@math.colostate.edu)
VALÉRY MAHÉ
Affiliation:
EPF Lausanne, SB-IMB-CSAG, Station 8, CH-1015 Lausanne, Switzerland (email: valery.mahe@epfl.ch)
JOSEPH H. SILVERMAN*
Affiliation:
Mathematics Department, Brown University, Box 1917, Providence, RI 02912, USA (email: jhs@math.brown.edu)
KATHERINE E. STANGE
Affiliation:
Department of Mathematics, Stanford University, 450 Serra Mall, Building 380, Stanford, CA 94305, USA (email: stange@math.stanford.edu)
MARCO STRENG
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (email: marco.streng@gmail.com)
*
✉For correspondence; e-mail: jhs@math.brown.edu
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Abstract

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In this note we study the existence of primes and of primitive divisors in function field analogues of classical divisibility sequences. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.

Keywords

lucas sequenceelliptic divisibility sequenceprimitive divisorfunction field over number field

MSC classification

Secondary: 11B39: Fibonacci and Lucas numbers and polynomials and generalizations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 92 , Issue 1 , February 2012 , pp. 99 - 126
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

Ingram’s research was supported by a grant from NSERC of Canada. Mahé’s research was supported by the Université de Franche-Comté. Silverman’s research was supported by DMS-0854755. Stange’s research was supported by NSERC PDF-373333 and NSF MSPRF 0802915. Streng’s research was supported by EPSRC grant no. EP/G004870/1.

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