Details

Roberto Dvornicich ^[a], Francesco Veneziano ^[b] and Umberto Zannier ^[c]

On the integral values of a curious recurrence

Pages: 1-18
Received: 24 February 2021
Accepted in revised form: 7 April 2021
Mathematics Subject Classification: 11B83, 11B99, 11D99.
Keywords: Mathematical Olympiad, Recurrence relation, Integral values.
Authors address:
[a]: Dipartimento di Matematica, Università di Pisa, Largo Pontecorvo 5, 56127 Pisa, Italia
[b]: Department of Mathematics, University of Genova, Via Dodecaneso 35, 16146 Genova, Italy
[c]: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italia

Abstract: We discuss an elementary problem, initially proposed for the Romanian Mathematical Olympiad, which leads to interesting remarks of various nature. We relate the problem to the theory of linear recurrence sequences with non-constant coefficients and their $p$-adic behaviour. Our considerations can be applied to a larger set of similarly-defined recurrences.

References

[BT33]: G. D. Birkhoff and W. J. Trjitzinsky, Analytic theory of singular difference equations, Acta Math. 60 (1933), 1-89. MR1555364 | DOI
[DGS94]: B. Dwork, G. Gerotto and F. J. Sullivan, An introduction to G-functions, Ann. of Math. Stud., 133, Princeton University Press, Princeton, NJ, 1994. MR1274045 | JSTOR
[MW56]: L. Moser and M. Wyman, Asymptotic expansions, Canadian J. Math. 8 (1956), 225-233. MR0078488 | DOI
[OEIS]: The OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, 2020, https://oeis.org
[RMS]: Romanian Mathematical Society, RMC 2010, Paralela 45 Publishing House, Bucharest, 2010.
[Wil06]: H. S. Wilf, Generatingfunctionology, 3rd ed., A K Peters, Wellesley, MA, 2006. MR2172781
[WZ85]: J. Wimp and D. Zeilberger, Resurrecting the asymptotics of linear recurrences, J. Math. Anal. Appl. 111 (1985), 162-176. MR0808671 | DOI