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A proof of the Hoggatt-Bergum conjecture. (English) Zbl 0937.11011

A set of positive integers \( \{a_{1}, a_{2}, \ldots , a_{m}\}\) is called a diophantine \(m\)-tuple (in the narrow sense) if \(a_{i}a_{j}+1\) is a perfect square for all \(1 \leq i < j \leq m\). The first examples of diophantine quadruples \( \{a, b, s, d \}\) were constructed by Fermat and Euler [see L. E. Dickson, History of the Theory of Numbers, Vol. 2, Chelsea, New York, pp. 523-529 (1966)]. The well-known result of V. E. Hoggatt and G. E. Bergum [Fibonacci Q. 15, 323-330 (1977; Zbl 0383.10007)] implies that for any \(k \geq 1\) the four integers \(F_{2k}\), \(F_{2k+2}\), \(F_{2k+4}\) and \(d= 4 F_{2k+1} F_{2k+2} F_{2k+3}\), where \(F_{n}\) is the \(n\)-th Fibonacci number, form a diophantine quadruple. Hoggatt and Bergum conjectured that the value \(d\) in any diophantine quadruple of the form \(\{F_{2k},F_{2k+2}, F_{2k+4}, d \}\) is unique. This conjecture for \(k=1\) was first proved by A. Baker and H. M. Davenport [Q. J. Math., Oxf. (2) 20, 129-137 (1969; Zbl 0177.06802)].
The author extends the Baker-Davenport result and proves the Hoggatt-Bergum conjecture for all positive integers \(k\).

MSC:

11D09 Quadratic and bilinear Diophantine equations
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11D25 Cubic and quartic Diophantine equations
11J86 Linear forms in logarithms; Baker’s method
11Y50 Computer solution of Diophantine equations
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