On the Diophantine Equation z(n) = (2 − 1/k)n Involving the Order of Appearance in the Fibonacci Sequence
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Let ( F n ) n ≥ 0 be the sequence of the Fibonacci numbers. The order (or rank) of appearance z ( n ) of a positive integer n is defined as the smallest positive integer m such that n divides F m . In 1975, Sallé proved that z ( n ) ≤ 2 n , for all positive integers n. In this paper, we shall solve the Diophantine equation z ( n ) = ( 2 − 1 / k ) n for positive integers n and k.
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2013 ◽
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pp. 316-321
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2010 ◽
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pp. 177-185
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pp. 813-821
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pp. 9-19
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pp. 50-105
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pp. 1117-1128
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