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Proceedings of the American Mathematical Society Series B

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society Series B (BPROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 2330-1511

The 2020 MCQ for Proceedings of the American Mathematical Society Series B is 0.84.

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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
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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