Repdigits base b as products of two Fibonacci numbers

Fatih Erduvan (), Refik Keskin () and Zafer Şiar ()
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Fatih Erduvan: Sakarya University
Refik Keskin: Sakarya University
Zafer Şiar: Bingöl University

Indian Journal of Pure and Applied Mathematics, 2021, vol. 52, issue 3, 861-868

Abstract: Abstract Let $$(F_{n})$$ ( F n ) be the sequence of Fibonacci numbers defined by $$F_{0}=0,~F_{1}=1$$ F 0 = 0 , F 1 = 1 , and $$F_{n}=F_{n-1}+F_{n-2}$$ F n = F n - 1 + F n - 2 for $$n\ge 2.$$ n ≥ 2 . Let $$2\le m\le n$$ 2 ≤ m ≤ n and $$b=2,3,4,5,6,7,8,9.$$ b = 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . In this study, we show that if $$F_{m}F_{n}$$ F m F n is a repdigit in base b and has at least two digits, then $$\begin{aligned} F_{m}F_{n}\in \left\{ 3,4,5,6,8,9,10,13,15,16,21,24,26,40,42,63,170,273\right\} . \end{aligned}$$ F m F n ∈ 3 , 4 , 5 , 6 , 8 , 9 , 10 , 13 , 15 , 16 , 21 , 24 , 26 , 40 , 42 , 63 , 170 , 273 . Furthermore, it is shown that if $$F_{n}$$ F n is a repdigit in base b and has at least two digits, then $$\begin{aligned} (n,b)=(7,3),(8,4),(8,6),(4,2),(5,4),(6,3),(6,7). \end{aligned}$$ ( n , b ) = ( 7 , 3 ) , ( 8 , 4 ) , ( 8 , 6 ) , ( 4 , 2 ) , ( 5 , 4 ) , ( 6 , 3 ) , ( 6 , 7 ) .

Keywords: Fibonacci number; Repdigit; Diophantine equations; Linear forms in logarithms; Baker’s Theory; 11B39; 11J86; 11D61 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s13226-021-00041-8

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