Abstract
In this paper, we look at the problem of expressing a term of a given nondegenerate binary recurrence sequence as a linear combination of a factorial and an \(S\)-unit whose coefficients are bounded. In particular, we find the largest member of the Fibonacci sequence which can be written as a sum or a difference between a factorial and an \(S\)-unit associated to the set of primes \(\{2,3,5,7\}\).
Résumé
Dans cet article, nous étudions le problème de comment représenter un terme d’une suite récurrente binaire non dégénérée comme une combinaison linéaire d’une factorielle et d’une \(S\)-unité dont les coefficients sont bornés. En particulier, nous trouvons le plus grand nombre de Fibonacci qui s’écrit comme la somme ou la différence d’une factorielle et d’une \(S\)-unité associée à l’ensemble des premiers \(\{2,3,5,7\}\).
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Acknowledgments
We thank the referee for comments which improved the quality of this paper. The preparation of this paper, F. Luca was supported in part by Project PAPIIT.
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Sanchez, S.G., Luca, F. Linear combinations of factorials and \(S\)-units in a binary recurrence sequence. Ann. Math. Québec 38, 169–188 (2014). https://doi.org/10.1007/s40316-014-0025-z
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DOI: https://doi.org/10.1007/s40316-014-0025-z