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Balancing Diophantine triples with distance 1. (English) Zbl 1374.11063

Let \(\{B_n\}_{n\geq 0}\) denote the sequence of Balancing numbers given by \(B_0 = 0\), \(B_1 = 1\) and \(B_{n+2} = 6B_{n+1} - B_n\) for all integers \(n\geq 0\). For a positive real number \(w\) we define the Balancing distance of \(w\) by \(\| w\|_B = \min\{| w - B_n | : n \geq 0\}\). This notion is analogous to the Fibonacci distance \(\| w\|_F\) introduced by F. Luca. They showed that if \(1 \leq a < b < c\) are integers, then \[ \max\{\| ab\|_F, \| ac\|_F, \| bc\|_F \} > \exp(0.034\sqrt{\log c}). \] This result has a numerical corollary, namely if \[ \max\{\| ab\|_F, \| ac\|_F, \| bc\|_F\} \leq 2, \] then \(c \leq \exp(415.62)\). In fact, the solution with maximal \(c\) to the inequality above is \[ (a, b, c) = (1, 11, 235). \] Note, that the origin of the problem is to solve the system of Diophantine equations \(ab + 1 = G_x\), \(ac + 1 = G_y\) and \(bc + 1 = G_z\), where the sequence \(\{G\}^{\infty}_{n=0}\) satisfies a given recurrence relation of order two. The main result of this work is the following.
Theorem Suppose that \(\varepsilon_x\), \(\varepsilon_y\) and \(\varepsilon_z\) are all in the set \(\{\pm 1\}\). If there exist integers \(1 \leq a < b < c\) such that \[ \begin{aligned} ab + \varepsilon_x &= B_x,\\ ac + \varepsilon_ y &= B_y,\\ bc + \varepsilon_z &= Bz \end{aligned} \] hold with positive integers \(x\), \(y\) and \(z\), then \((a, b, c) = (1, 34, 1188)\), \((x, y, z) = (3, 5, 7)\), \((\varepsilon_x, \varepsilon_y, \varepsilon_z) = (1, 1, -1)\).

MSC:

11D72 Diophantine equations in many variables
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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References:

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