A note on quasi-smoothly undulating numbers

Authors

DOI:

https://doi.org/10.22481/intermaths.v5i2.15564

Keywords:

Divisibility, Perfect Squares, Primes, Quasi-smoothly Undulating

Abstract

The smoothly oscillating numbers can be understood as specific instances of quasismoothly oscillating integers. While some studies present prime numbers within the class of smoothly oscillating numbers and demonstrate various divisibility criteria, it has been shown that, for smoothly oscillating numbers of the \(one-zero\) type, there is only one prime number, which is 101. Additionally, other research illustrates connections between smoothly oscillating numbers and powers with natural exponents. In this paper, we present the results of our investigation into the new class of integers known as quasismoothly oscillating numbers. We explore properties related to divisibility criteria, primality, and perfect squares, analyzing some subclasses of quasismoothly oscillating numbers. We provide expressions for their determination and partial generating sums for this class of numbers. The tools and properties employed in the proofs are elementary, primarily involving concepts of divisibility and congruence.

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Author Biographies

Eudes Antonio Costa, Universidade Federal do Tocantins, Arraias - TO, Brasil

Adjunct Professor at the Federal University of Tocantins, Arraias Campus (Mathematics Course). Post-doctorate in Mathematics from the Federal University of Ceará (2019), PhD in Mathematics from the University of Brasília (2013), Master's degree in Mathematics from the Federal University of Goiás (2001), undergraduate degree in Mathematics from the Federal University of Goiás (1998) and undergraduate degree in Philosophy from the Pontifical Catholic University of Goiás (1995). Experience with Teacher Training (PROFMAT, Bachelor's Degree Course and Improvement Courses) and Mathematics Olympiads (OBM and OBMEP).

Fernando Soares Carvalho , Universidade Federal do Tocantins, Arraias - TO, Brasil

PhD in Mechanical Sciences - Faculty of Technology / UnB (2020). Master's in Mathematics from the Federal University of Goiás (2011). Specialization in Mathematics and Statistics from the Federal University of Lavras - MG (2009). Holds a bachelor's degree in Mathematics from the Federal University of Goiás (2003). Was a tenured professor at the State University of Goiás (2010), Faculdade de Ceres (FACERES) (2008-2010), the Federal University of Goiás (2005-2008), and the public school system of the State of Goiás (2004-2010). He is currently a tenured associate professor at the Federal University and a permanent faculty member of the Professional Master's Program in Mathematics (PROFMAT). He has been an evaluator of undergraduate programs (in-person and distance learning) for INEP/MEC since 2018. Member of the research groups: Group of Experimental and Computational Mechanics - GMEC/UnB and Group of Studies and Research in Mathematics and Mathematics Education - UFT. He has experience in undergraduate and graduate courses in Mathematics, with an emphasis on: Teaching Mathematics for pre-service mathematics teachers. His research focuses on Mathematical and Computational Modeling, Mathematics Education, and Applied Mathematics in Engineering: Topological Sensitivity Analysis, Topological Derivative and Topology Optimization.

References

F. S. Carvalho; E.A. Costa. “Um passeio pelos números ondulantes”. REMAT: Revista Eletrônica da Matemática, vol. 8, no. 2, p. e3001-e3001, 2022.

E. A. Costa; A. B. Souza. “N´umeros ondulantes na forma 101...01”. Gazeta Matemática , SPM, 2024.

C. A. Pickover . Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning . (Chapters 52 and 88). Oxford University Press, 2003.345

D. C. Santos; E. A. Costa. “Peculiarities of smoothly undulating number”. Intermaths ,vol. 4(2), pp. 38-53., 2023.

N. J. A. Sloane. 0EIS - The on-line encyclopedia of integer sequences, http://oeis.org/A002275 .

PUTNAM Problems. 50th Putnam 1989. Disponível em: https://prase.cz/kalva/putnam.html. Acesso em: 9 jan. 2024.

A. Hefez. Aritm´etica . SBM-Coleção PROFMAT, 2a. ed. Rio de Janeiro-RJ. SBM, 2016.

E. A. Costa; D. C. Santos. “Os n´umeros suavemente ondulantes generalizados”. REMAT: Revista Eletrônica da Matemática, 10(2), e3008-e3008, 2024.

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Published

2024-12-31

How to Cite

Santos, D. C., Costa, E. A., & Carvalho , F. S. (2024). A note on quasi-smoothly undulating numbers. Intermaths, 5(2), 93-105. https://doi.org/10.22481/intermaths.v5i2.15564

Issue

Vol. 5 No. 2 (2024)

Section

Artigos

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