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A linear recurrence sequence of composite numbers

Published online by Cambridge University Press:  01 November 2012

Jonas Šiurys*
Affiliation:
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania (email: [email protected])

Abstract

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We prove that for each positive integer k in the range 2≤k≤10 and for each positive integer k≡79 (mod 120) there is a k-step Fibonacci-like sequence of composite numbers and give some examples of such sequences. This is a natural extension of a result of Graham for the Fibonacci-like sequence.

Type
Research Article
Copyright
Copyright © London Mathematical Society 2012

References

[1]Bilu, Y., Hanrot, G., Voutier, P. M. and Mignotte, M., ‘Existence of primitive divisors of Lucas and Lehmer numbers’, J. reine angew. Math. 539 (2001) 75122.Google Scholar
[2]Choi, S. L. G., ‘Covering the set of integers by congruence classes of distinct moduli’, Math. Comp. 25 (1971) 885895.CrossRefGoogle Scholar
[3]Dubickas, A., Novikas, A. and Šiurys, J., ‘A binary linear recurrence sequence of composite numbers’, J. Number Theory 130 (2010) 17371749.Google Scholar
[4]Flores, I., ‘Direct calculation of k-generalized Fibonacci numbers’, Fibonacci Quart. 5 (1967) 259266.Google Scholar
[5]Graham, R. L., ‘A Fibonacci-like sequence of composite numbers’, Math. Mag. 37 (1964) 322324.CrossRefGoogle Scholar
[6]Hall, M., ‘Divisibility sequences of third order’, Amer. J. Math. 58 (1936) 577584.CrossRefGoogle Scholar
[7]Knuth, D. E., ‘A Fibonacci-like sequence of composite numbers’, Math. Mag. 63 (1990) 2125.CrossRefGoogle Scholar
[8]Nicol, J. W., ‘A Fibonacci-like sequence of composite numbers’, Electron. J. Combin. 6 (1999) research paper R44.CrossRefGoogle Scholar
[9]Noe, T. D. and Post, J. V., ‘Primes in Fibonacci n-step and Lucas n-step sequences’, J. Integer Seq. 8 (2005) article 05.4.4.Google Scholar
[10] The PARI Group, ‘Bordeaux. PARI/GP, version, 2.3.5’, 2006, http://pari.math.u-bordeaux.fr/.Google Scholar
[11]Parnami, J. C. and Shorey, T. N., ‘Subsequences of binary recursive sequences’, Acta Arith. 40 (1982) 193196.Google Scholar
[12]Somer, L., ‘Second-order linear recurrences of composite numbers’, Fibonacci Quart. 44 (2006) 358361.Google Scholar
[13]Šiurys, J., ‘A tribonacci-like sequence of composite numbers’, Fibonacci Quart. 49 (2011) 298302.Google Scholar
[14]Vsemirnov, M., ‘A new Fibonacci-like sequence of composite numbers’, J. Integer Seq. 7 (2004) article 04.3.7.Google Scholar
[15]Wilf, H. S., ‘Letters to the editor’, Math. Mag. 63 (1990) 284.Google Scholar