## The density of primes dividing a term in the Somos-5 sequence

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Bryant Davis, Rebecca Kotsonis and Jeremy Rouse
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**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