A binary linear recurrence sequence of composite numbers |
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Authors: | Artūras Dubickas Aivaras Novikas |
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Institution: | Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania |
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Abstract: | Let (a,b)∈Z2, where b≠0 and (a,b)≠(±2,−1). We prove that then there exist two positive relatively prime composite integers x1, x2 such that the sequence given by xn+1=axn+bxn−1, n=2,3,… , consists of composite terms only, i.e., |xn| is a composite integer for each n∈N. In the proof of this result we use certain covering systems, divisibility sequences and, for some special pairs (a,±1), computer calculations. The paper is motivated by a result of Graham who proved this theorem in the special case of the Fibonacci-like sequence, where (a,b)=(1,1). |
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Keywords: | 11B37 11A07 11Y55 |
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