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Balancing Diophantine triples. (English) Zbl 1293.11050

A Diophantine \(m\)-tuple is a set \(\{a_1,\dots,a_m\}\) of positive integers such that \(a_ia_j+1\) is square for all \(1\leq i<j\leq m\). Diophantus investigated first the problem of finding rational quadruples, and he provided one example: \(\frac{1}{16}\), \(\frac{33}{16}\), \(\frac{68}{16}\), \(\frac{105}{16}\). The first integer quadruple, \(\{1,3,8,120\}\) was found by Fermat. Infinitely many Diophantine quadruples of integers are known and it is conjectured that there is no integer Diophantine quintuple. This was almost proved by A. Dujella, who showed that there can be at most finitely many Diophantine quintuples and all of them are, at least in theory, effectively computable.
The first definition of balancing numbers is essentially due to R. P. Finkelstein, although he called them numerical centers. A positive integer \(n\) is called balancing number if \[ 1+2+\cdots+(n-1)=(n+1)+(n+2)+\cdots+(n+r) \] holds for some positive integer \(r\). Then \(r\) is called balancer corresponding to the balancing number \(n\).
It is well known that there are at most finitely many Fibonacci and Lucas Diophantine triples, but there wasn’t a hint to find all of them. F. Luca and L. Szalay described a method to determine Diophantine triples for Fibonacci numbers and Lucas numbers. In this paper, the authors follow their method, although some new types of problems appeared when they proved the following:
Theorem. There do no exist positive integers \(a<b<c\) such that \[ ab+1=B_x,\qquad ac+1=B_y,\qquad bc+1=B_z, \] \(0<x<y<z\) are natural numbers and \((B_n)_{n=0}^\infty\) is the sequence of balancing numbers.
For more details see this paper.

MSC:

11D09 Quadratic and bilinear Diophantine equations
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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