Fibonacci and Lucas numbers with only one distinct digit. (English) Zbl 0958.11007

Let \(F_n\) and \(L_n\) denote respectively the Fibonacci and Lucas numbers. The main results of this paper are: Theorem. If \(F_n=a(10^m-1)/9\) [respectively, \(L_n=a(10^m-1)/9\)] for some integer \(0\leq a\leq 9\), then \(0\leq n\leq 6\) or \(n=10\) [respectively, \(0\leq n\leq 5\)]. Surprisingly, the methods of proofs are completely elementary. The author makes no use of Baker’s theory of linear forms in logarithms, he just uses elementary algebra, congruences and the quadratic reciprocity law in a very clever way.


11A63 Radix representation; digital problems
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11D61 Exponential Diophantine equations
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