The density of primes dividing a term in the Somos-5 sequence
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- by Bryant Davis, Rebecca Kotsonis and Jeremy Rouse HTML | PDF
- Proc. Amer. Math. Soc. Ser. B 4 (2017), 5-20
Abstract:
The Somos-5 sequence is defined by $a_{0} = a_{1} = a_{2} = a_{3} = a_{4} = 1$ and $a_{m} = \frac {a_{m-1} a_{m-4} + a_{m-2} a_{m-3}}{a_{m-5}}$ for $m \geq 5$. We relate the arithmetic of the Somos-5 sequence to the elliptic curve $E : y^{2} + xy = x^{3} + x^{2} - 2x$ and use properties of Galois representations attached to $E$ to prove the density of primes $p$ dividing some term in the Somos-5 sequence is equal to $\frac {5087}{10752}$.References
- Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no.ย 3-4, 235โ265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
- Paul Cubre and Jeremy Rouse, Divisibility properties of the Fibonacci entry point, Proc. Amer. Math. Soc. 142 (2014), no.ย 11, 3771โ3785. MR 3251719, DOI 10.1090/S0002-9939-2014-12269-6
- Sergey Fomin and Andrei Zelevinsky, The Laurent phenomenon, Adv. in Appl. Math. 28 (2002), no.ย 2, 119โ144. MR 1888840, DOI 10.1006/aama.2001.0770
- Helmut Hasse, รber die Dichte der Primzahlen $p$, fรผr die eine vorgegebene ganzrationale Zahl $a\not =0$ von gerader bzw. ungerader Ordnung $\textrm {mod}.p$ ist, Math. Ann. 166 (1966), 19โ23 (German). MR 205975, DOI 10.1007/BF01361432
- A. N. W. Hone, Elliptic curves and quadratic recurrence sequences, Bull. London Math. Soc. 37 (2005), no.ย 2, 161โ171. MR 2119015, DOI 10.1112/S0024609304004163
- A. N. W. Hone, Sigma function solution of the initial value problem for Somos 5 sequences, Trans. Amer. Math. Soc. 359 (2007), no.ย 10, 5019โ5034. MR 2320658, DOI 10.1090/S0002-9947-07-04215-8
- Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI 10.1090/coll/053
- Rafe Jones and Jeremy Rouse, Galois theory of iterated endomorphisms, Proc. Lond. Math. Soc. (3) 100 (2010), no.ย 3, 763โ794. Appendix A by Jeffrey D. Achter. MR 2640290, DOI 10.1112/plms/pdp051
- J. C. Lagarias, The set of primes dividing the Lucas numbers has density $2/3$, Pacific J. Math. 118 (1985), no.ย 2, 449โ461. MR 789184, DOI 10.2140/pjm.1985.118.449
- J. C. Lagarias, Errata to: โThe set of primes dividing the Lucas numbers has density $2/3$โ [Pacific J. Math. 118 (1985), no. 2, 449โ461; MR0789184 (86i:11007)], Pacific J. Math. 162 (1994), no.ย 2, 393โ396. MR 1251907, DOI 10.2140/pjm.1994.162.393
- Richard Pink, On the order of the reduction of a point on an abelian variety, Math. Ann. 330 (2004), no.ย 2, 275โ291. MR 2089426, DOI 10.1007/s00208-004-0548-8
- Jeremy Rouse and David Zureick-Brown, Elliptic curves over $\Bbb Q$ and 2-adic images of Galois, Res. Number Theory 1 (2015), Paper No. 12, 34. MR 3500996, DOI 10.1007/s40993-015-0013-7
- Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1992. Corrected reprint of the 1986 original. MR 1329092, DOI 10.1007/978-1-4757-4252-7
- Joseph H. Silverman and John Tate, Rational points on elliptic curves, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. MR 1171452, DOI 10.1007/978-1-4757-4252-7
- David E. Speyer, Perfect matchings and the octahedron recurrence, J. Algebraic Combin. 25 (2007), no.ย 3, 309โ348. MR 2317336, DOI 10.1007/s10801-006-0039-y
Additional Information
- Bryant Davis
- Affiliation: Department of Mathematics and Statistics, Wake Forest University, Winston-Salem, North Carolina 27109
- Address at time of publication: Department of Statistics, University of Florida, Gainesville, Florida 32611
- Email: davibf11@ufl.edu
- Rebecca Kotsonis
- Affiliation: Department of Mathematics and Statistics, Wake Forest University, Winston-Salem, North Carolina 27109
- Email: rkotsonis@uchicago.edu
- Jeremy Rouse
- Affiliation: Department of Mathematics and Statistics, Wake Forest University, Winston-Salem, North Carolina 27109
- MR Author ID: 741123
- Email: rouseja@wfu.edu
- Received by editor(s): July 21, 2015
- Received by editor(s) in revised form: August 26, 2016
- Published electronically: August 3, 2017
- Communicated by: Matthew A. Papanikolas
- © Copyright 2017 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- Journal: Proc. Amer. Math. Soc. Ser. B 4 (2017), 5-20
- MSC (2010): Primary 11G05; Secondary 11F80
- DOI: https://doi.org/10.1090/bproc/26
- MathSciNet review: 3681974