On Fibonacci Numbers of Order r Which Are Expressible as Sum of Consecutive Factorial Numbers

Eva Trojovská and Pavel Trojovský
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Eva Trojovská: Department of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic
Pavel Trojovský: Department of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic

Mathematics, 2021, vol. 9, issue 9, 1-9

Abstract: Let ( t n ( r ) ) n ? 0 be the sequence of the generalized Fibonacci number of order r , which is defined by the recurrence t n ( r ) = t n ? 1 ( r ) + ? + t n ? r ( r ) for n ? r , with initial values t 0 ( r ) = 0 and t i ( r ) = 1 , for all 1 ? i ? r . In 2002, Grossman and Luca searched for terms of the sequence ( t n ( 2 ) ) n , which are expressible as a sum of factorials. In this paper, we continue this program by proving that, for any ? ? 1 , there exists an effectively computable constant C = C ( ? ) > 0 (only depending on ? ), such that, if ( m , n , r ) is a solution of t m ( r ) = n ! + ( n + 1 ) ! + ? + ( n + ? ) ! , with r even, then max { m , n , r } < C . As an application, we solve the previous equation for all 1 ? ? ? 5 .

Keywords: diophantine equation; factorial; fibonacci r-numbers; 2-adic valuation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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