On the exponential Diophantine equation $P_n^x+P_{n+1}^x=P_m$

On the exponential Diophantine equation $P_n^x+P_{n+1}^x=P_m$

In this paper, we find all the solutions of the title Diophantine equation in nonnegative integer variables(m, n, x) , where $P_k$ is the k th term of the Pell sequence.

___

  • [1] Baker A, Davenport H. The equations 3x2 ? 2 = y2 and 8x2 ? 7 = z2 . Quarterly Journal of Mathematics 1969; 20: 129-137.
  • [2] Bertók C, Hajdu L, Pink I, Rábai Z. Linear combinations of prime powers in binary recurrence sequences. International Journal of Number Theory 2017; 13: 261-271.
  • [3] Bravo JJ, Luca F. Coincidences in generalized Fibonacci recurrences. Journal of Number Theory 2013; 133: 2121-2137.
  • [4] Dujella A, Pethő A. A generalization of a theorem of Baker and Davenport. Quarterly Journal of Mathematics 1998; 49 (2): 291-306.
  • [5] Luca F, Oyono R. An exponential Diophantine equation related to powers of two consecutive Fibonacci numbers. Proceedings of the Japan Academy Series A 2011; 87: 45-50.
  • [6] Marques D, Togbé A. On the sum of powers of two consecutive Fibonacci numbers. Proceedings of the Japan Academy Series A 2010; 86: 174-176.
  • [7] Matveev EM. An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers, II. Izvestiya Mathematics 2000; 64: 1217-1269.
  • [8] Ruiz CAG, Luca F. An exponential Diophantine equation related to the sum of powers of two consecutive k - generalized Fibonacci numbers. College Mathematics Journal 2014; 137: 171-188.
  • [9] Shorey TN, Tijdeman R. Exponential Diophantine Equations. Cambridge, UK: Cambridge University Press, 1986.