A Quadratic Diophantine Equation Involving Generalized Fibonacci Numbers

Ana Paula Chaves and Pavel Trojovský
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Ana Paula Chaves: Instituto de Matemática e estatística, Universidade federal de Goiás, Goiás 74690-900, Brazil
Pavel Trojovský: Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic

Mathematics, 2020, vol. 8, issue 6, 1-10

Abstract: The sequence of the k -generalized Fibonacci numbers ( F n ( k ) ) n is defined by the recurrence F n ( k ) = ∑ j = 1 k F n − j ( k ) beginning with the k terms 0 , … , 0 , 1 . In this paper, we shall solve the Diophantine equation F n ( k ) = ( F m ( l ) ) 2 + 1 , in positive integers m , n , k and l .

Keywords: Fibonacci number; recurrence sequence; linear form in logarithms; reduction method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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