On some Diophantine equations involving balancing numbers

Euloge Tchammou; Alain Togbé

Archivum Mathematicum (2021)

  • Volume: 057, Issue: 2, page 113-130
  • ISSN: 0044-8753

Abstract

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In this paper, we find all the solutions of the Diophantine equation B 1 p + 2 B 2 p + ⋯ + k B k p = B n q in positive integer variables ( k , n ) , where B i is the i t h balancing number if the exponents p , q are included in the set { 1 , 2 } .

How to cite

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Tchammou, Euloge, and Togbé, Alain. "On some Diophantine equations involving balancing numbers." Archivum Mathematicum 057.2 (2021): 113-130. <http://eudml.org/doc/297477>.

@article{Tchammou2021,
abstract = {In this paper, we find all the solutions of the Diophantine equation $B_1^p+2B_2^p+\cdots +kB_k^p=B_n^q$ in positive integer variables $(k, n)$, where $B_i$ is the $i^\{th\}$ balancing number if the exponents $p$, $ q$ are included in the set $\lbrace 1,2\rbrace $.},
author = {Tchammou, Euloge, Togbé, Alain},
journal = {Archivum Mathematicum},
keywords = {balancing numbers; Pell numbers; Diophantine equation},
language = {eng},
number = {2},
pages = {113-130},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On some Diophantine equations involving balancing numbers},
url = {http://eudml.org/doc/297477},
volume = {057},
year = {2021},
}

TY - JOUR
AU - Tchammou, Euloge
AU - Togbé, Alain
TI - On some Diophantine equations involving balancing numbers
JO - Archivum Mathematicum
PY - 2021
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 057
IS - 2
SP - 113
EP - 130
AB - In this paper, we find all the solutions of the Diophantine equation $B_1^p+2B_2^p+\cdots +kB_k^p=B_n^q$ in positive integer variables $(k, n)$, where $B_i$ is the $i^{th}$ balancing number if the exponents $p$, $ q$ are included in the set $\lbrace 1,2\rbrace $.
LA - eng
KW - balancing numbers; Pell numbers; Diophantine equation
UR - http://eudml.org/doc/297477
ER -

References

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  1. Altassan, A., Luca, F., On the Diophantine equation ∑ j = 1 k j F j p = F n q , J. Number Theory 217 (2020), 256–277. (2020) MR4140628
  2. Behera, A., Panda, G.K., On the square roots of triangular numbers, Fibonacci Quart. 37 (2) (1999), 98–105. (1999) MR1690458
  3. Catarino, P., Campos, H., Vasco, P., On some identities for balancing and cobalancing numbers, Ann. Math. Inform. 45 (2015), 11–24. (2015) MR3438809
  4. Gueth, K., Luca, F., Szalay, L., Solutions to F 1 p + 2 F 2 p + ⋯ + k F k p = F n q with small given exponents, Proc. Japan Acad. Ser. A, Math. Sci. 96 (4) (2020), 33–37. (2020) MR4080788
  5. Horadam, A.F., Pell identities, Fibonacci Quart. 9 (3) (1971), 245–263. (1971) MR0308029
  6. Niven, I., Zuckerman, H.S., Montgomery, H.L., An Introduction to the Theory of Numbers, fifth edition ed., John Wiley & Sons, Inc., New York, 1991. (1991) MR1083765
  7. Olajos, P., Properties of Balancing, Cobalancing and Generalized Balancing Numbers, Ann. Math. Inform. 37 (2010), 125–138. (2010) MR2753032
  8. Panda, G.K., Some Fascinating Properties of Balancing Numbers, Proceedings of the Eleventh International Conference on Fibonacci Numbers and their Applications, Cong. Numer., vol. 194, 2009, pp. 185–189. (2009) MR2463534
  9. Panda, G.K., Ray, P.K., Some links of balancing and cobalancing numbers with Pell and associated Pell numbers, Bul. Inst. Math. Acad. Sinica 6 (2011), 41–72. (2011) MR2850087
  10. Ray, P.K., Balancing and cobalancing numbers, Ph.D. thesis, National Institute of Technology, Rourkela, India, 2009. (2009) 
  11. Soydan, G., Németh, L., Szalay, L., On the diophantine equation ∑ j = 1 k j F j p = F n q , Arch. Math. (Brno) 54 (2008), 177–188. (2008) MR3847324
  12. Tchammou, E., Togbé, A., 10.1007/s10474-020-01043-4, Acta Math. Hungar. 162 (2) (2020), 647–676. (2020) MR4173320DOI10.1007/s10474-020-01043-4

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