On k-Pell numbers close to power of 2

• Autores: Mohamadou Bachabi, Alain Togbé
• Localización: Revista Colombiana de Matemáticas, ISSN-e 0034-7426, Vol. 58, Nº. 1, 2024, págs. 67-80
• Idioma: varios idiomas
• Títulos paralelos:
  • Números de k-Pell cercanos a potencias de 2
• Enlaces
  • Texto completo
• Resumen

  • For k ≥ 2, let (P(k)n)n≥2−k be the k-generalized Pell sequence which starts with 0, · · · , 0, 1 (k terms) and each term afterwards is given by the linear recurrence P(k)n = 2P(k)n-1 + P(k)n-2 + · · · + P(k)n-k, for all n ≥ 2.

    An integer n is said to be close to a positive integer m if n satisfies |n−m| < √m. In this paper, we solve the Diophantine inequality |P(k)n - 2m| < 2m/2, in positive unknowns k, n, and m.

• Referencias bibliográficas
  • [1] A. A¸cikel, N. Irmak, and L. Szalay, The k-generalized Lucas numbers close to a power of 2, To appear in Mathematica Slovaca.
  • [2] A. Baker and H. Davenport, The equations 3x2−2 = y2 and 8x2−7 = z2, Quart.J of Math. Ser. 20 (1969), no. 2, 129-137.
  • [3] J. J. Bravo, C. A. Gómez, and J. L. Herrera, k-fibonacci numbers close to a power of 2, Quaest. Math. 44 (2021), no. 12, 1681-1690.
  • [4] J. J. Bravo, C. A. Gómez, and F. Luca, Power of two as sums of two k-Fibonacci numbers, Miskolc Math. Notes 17 (2016), 85-100.
  • [5] J. J. Bravo and J. L. Herrera, Repdigits in generalized Pell sequences, Archivum Mathematicum 56 (2020), no. 4, 249-262.
  • [6] J. J. Bravo, J. L. Herrera, and F. Luca, On a generalization of the Pell sequence, Math. Bohema. 146 (2021), no. 2, 199-213.
  • [7] J. J. Bravo and F. Luca, Powers of two in generalized Fibonacci sequences, Rev. Colombiana Mat. 46 (2012), no. 1, 67-79.
  • [8] Y. Bugeaud, M. Maurice, and S. Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect...
  • [9] E. Kili¸c, On the usual Fibonacci and generalized order-k Pell numbers, Ars Combin 109 (2013), 391-403.
  • [10] S. Chern and A. Cui, Fibonacci numbers close to a power of 2, Fibonacci Quart. 52 (2014), no. 4, 344-348.
  • [11] A. Dujella and A. Pethö, A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. 49 (1998), no. 2, 291-306.
  • [12] E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers II, Izv. Ross. Akad....
  • [13] B. V. Normenyo, S. E. Rihane, and A. Togbé, Common terms of k-Pell numbers and Padovan or Perrin numbers, Arab. J. Math. 12 (2023), 219-232.
  • [14] S. Guzmán Sánchez and F. Luca, Linear combinations of factorials and S-units in a binary recurrence sequence, Ann. Math. Québec 38 (2014),...
  • [15] M. Waldshmidt, Diophantine approximation on linear algebraic groups: transcendence properties of the exponential function in several...