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 \(k\).


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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