Fibonacci and Lucas numbers as products of two repdigits

Fibonacci and Lucas numbers as products of two repdigits

In this study, it is shown that the largest Fibonacci number that is the product of two repdigits is $F_{10}=55=5.11=55.1$ and the largest Lucas number that is the product of two repdigits is $L_6=18=2.9=3.6$.

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  • [1] Adegbindin C, Luca F, Togbé A. Pell and Pell-Lucas numbers as sums of two repdigits. Bulletin of the Malaysian Mathematical Sciences Society (in press). doi: 10.1007/s40840-019-00739-3
  • [2] Alvarado SD, Luca F. Fibonacci numbers which are sums of two repdigits. Aportaciones Matemáticas Investigación 2011; 20: 97-108.
  • [3] Baker A, Davenport H. The equations 3x2 − 2 = y2 and 8x2 − 7 = z2. Quarterly Journal of Mathematics Oxford Series 1969; 20 (1): 129-137.
  • [4] Bugeaud Y, Mignotte M, Siksek S. Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers. Annals of Mathematics 2006; 163 (3): 969-1018.
  • [5] Ddamulira M, Luca F, Rakotomalala M. Fibonacci numbers which are products of two Pell numbers. Fibonacci Quarterly 2016; 54 (1): 11-18.
  • [6] de Weger BMM. Algorithms for Diophantine equations. PhD, Eindhoven University of Technology, Eindhoven, the Netherlands, 1989.
  • [7] Debnath L. A short history of the Fibonacci and golden numbers with their applications. International Journal of Mathematical Education in Science and Technology 2011; 42 (3): 337-367.
  • [8] Dujella A, Pethò A. A generalization of a theorem of Baker and Davenport. Quarterly Journal of Mathematics and Oxford Series 1998; 49 (3): 291-306.
  • [9] Faye B, Luca F. Pell and Pell-Lucas numbers with only one distinct digit. Annales Mathematicae et Informaticae 2015; 45: 55-60.
  • [10] Luca F. Fibonacci and Lucas numbers with only one distinct digit. Portugaliae Mathematica 2000; 57 (2): 243-254.
  • [11] Matveev EM. An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II. Izvestiya Akademii Nauk Series Mathematics 2000; 64 (6): 125-180 (in Russian).