Repdigits as difference of two Fibonacci or Lucas numbers
Abstract
In the present study we investigate all repdigits which are expressed as a difference of two Fibonacci or Lucas numbers. We show that if Fn−Fm is a repdigit, where Fn denotes the n-th Fibonacci number, then (n,m)∈{(7,3),(9,1),(9,2),(11,1),(11,2), (11,9),(12,11),(15,10)}. Further, if Ln denotes the n-th Lucas number, then Ln−Lm is a repdigit for (n,m)∈{(6,4),(7,4),(7,6),(8,2)}, where n>m.
Namely, the only repdigits that can be expressed as difference of two Fibonacci numbers are 11,33,55,88 and 555; their representations are 11=F7−F3, 33=F9−F1=F9−F2, 55=F11−F9=F12−F11, 88=F11−F1=F11−F2, 555=F15−F10 (Theorem 2). Similar result for difference of two Lucas numbers: The only repdigits that can be expressed as difference of two Lucas numbers are 11,22 and 44; their representations are 11=L6−L4=L7−L6, 22=L7−L4, 4=L8−L2 (Theorem 3).
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