## 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

### Citations:

Zbl 0177.06802; Zbl 0383.10007
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