A note on repunit number sequence

Douglas Catulio Santos; Eudes Antonio Costa

doi:10.22481/intermaths.v5i1.14922

Autores

Douglas Catulio Santos https://orcid.org/0000-0002-5221-6087
Eudes Antonio Costa https://orcid.org/0000-0001-6684-9961

DOI:

https://doi.org/10.22481/intermaths.v5i1.14922

Resumo

In this paper, we investigate the classical identities of the repunit sequence with integer indices in light of the properties of Horadan-type sequences. We highlight particularly the Tagiuri-Vajda Identity and Gelin-Cesàro Identity. Additionally, we prove that no repunit is a perfect power, either even or odd. Finally, we address a divisibility criterion for the terms of repunit rn by a prime p and its powers.

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Referências

A. F. Horadam. “A generalized Fibonacci sequence." The American Mathematical Monthly,

v. 68, n. 5, p. 455–459, 1961.

A. F. Horadam. “Basic properties of a certain generalized sequence of numbers", The

Fibonacci Quart., v. 3, n. 3, p. 161–176, 1965.

G. Cerda. “Matrix methods in Horadam sequences". Boletín de Matemáticas, v. 19, n. 2, p.

–106, 2012.

N. J. A. Sloane. 0EIS - The on-line encyclopedia of integer sequences,

http://oeis.org/A002275.

D. C. Santos; E. A. Costa. “Um passeio pela sequência repunidade". CQD-Revista Eletrônica

Paulista de Matemática, p. 241-254, 2023. https://doi.org/10.21167/cqdv23n1ic2023241254

J. H. Jaroma. “Factoring Generalized Repunits". Bulletin of the Irish Mathematical Society,

n. 59, p. 29-35. 2007

D. Kalman; R. Mena. “The Fibonacci numbers-exposed". Mathematics magazine, v. 76, n.

, p. 167-181, 2003.

K. H. Rosen. Discrete mathematics and its applications. The McGraw Hill Companies,

A. H. Beiler. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains.

ed. Dover. 1966.

E. A. Costa; D. C. Santos. “Algumas propriedades dos números Monodígitos e Repunidades". Revista de Matemática da UFOP, v. 2, p. 47-58, 2022.

https://doi.org/10.5281/zenodo.11062161.

S. Yates. Repunits and repetends. Star Publishing Co., Inc. Boynton Beach, Florida, 1992.

E. A. Costa; D. C. Santos; F. S. Monteiro; V. M. A. Souza. “On the repunit sequence at negative indices". Revista de Matemática da UFOP, v. 1, p. 1-12, 2024.

https://doi.org/10.5281/zenodo.11062161.

MAOHUA, L. A note on perfect powers of the form x^{m−1} + ... + x + 1. Acta Arithmetica, v. 69, n. 1, p. 91–98, 1995.

MÜLER, T. Note on the diophantine equation 1 + 2^p + (2^p)^2 + ... + (2^p)^n = y^p. Elemente der Mathematik, v. 60, p. 148–149, 2005. Disponível em: https://ems.press/content/serial-article-files/7121. Acesso em: 23 mar. 2026.

S. Yates. “The Mystique of Repunits". Mathematics Magazine. v. 51, n. 1, p. 22-28, 1978.

https://doi.org/10.1080/0025570X.1978.11976671.

I. Niven; H.S. Zuckerman; H. L. Montegomery. An introduction to the theory of numbers .

John Wiley and Sons. 1991.