Nondefective integers with respect to certain Lucas sequences of the second kind.(English)Zbl 1412.11025
Summary: Let \(v(a, b)\) denote the Lucas sequence of the second kind defined by the second order recursion relation \(v_{n+2}= av_{n+1}+ bv_n\) with initial terms \(v_0= 2\) and \(v_1= a\), where \(a\) and \(b\) are integers. The positive integer \(m\) is said to be nondefective if \(v(a, b)\) contains a complete system of residues modulo \(m\). All possibilities for \(m\) to be nondefective are found when \(b = \pm 1\). This paper generalized results ofB. Avila andY. Chen [Fibonacci Q. 51, No. 2, 151–152 (2013;Zbl 1306.11016)] for the Lucas sequences \(\{L_n\}=v(1,1)\).
MSC:
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
Citations:
Zbl 1306.11016Cite
Full Text:Link
References:
| [1] | B. Avila and Y. Chen, On moduli for which the Fibonacci numbers contain a complete residue system, Fibonacci Quart. 51 (2013), 151-152. ·Zbl 1306.11016 |
| [2] | G. Bruckner, Fibonacci sequence modulo a prime p ⌘ 3 (mod 4), Fibonacci Quart. 8 (1970), 217-220. ·Zbl 0207.35202 |
| [3] | R. T. Bumby, A distribution property for linear recurrence of the second order, Proc. Amer. Math. Soc. 50 (1975), 101-106. ·Zbl 0318.10006 |
| [4] | S. A. Burr, On moduli for which the Fibonacci sequence contains a complete system of residues, Fibonacci Quart. 9 (1971), 497-504. ·Zbl 0227.10007 |
| [5] | R. D. Carmichael, On the numerical factors of arithmetic forms ↵n±n, Ann. of Math. 15 (1913), 30-70. ·JFM 44.0216.01 |
| [6] | R. D. Carmichael, On sequences of integers defined by recurrence relations, Quart. J. Pure Appl. Math. 48 (1920), 343-372. |
| [7] | D. H. Lehmer, An extended theory of Lucas’ functions, Ann. of Math. 31 (1930), 419-448. ·JFM 56.0874.04 |
| [8] | H.-C. Li, Complete and reduced systems of second-order recurrences modulo p, Fibonacci Quart. 38 (2000), 272-281. ·Zbl 0958.11020 |
| [9] | E. Lucas, Th´eorie des fonctions num´eriques simplement p´eriodiques, Amer. J. Math. 1 (1878), 184-240, 289-321. ·JFM 10.0134.05 |
| [10] | P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, New York, 1996. ·Zbl 0856.11001 |
| [11] | A. Schinzel, Special Lucas sequences including the Fibonacci sequence, modulo a prime, A Tribute to Paul Erd¨os, A. Baker et al. (eds.), Cambridge Univ. Press, Cambridge, 1990, 349-357. ·Zbl 0716.11009 |
| [12] | A. P. Shah, Fibonacci sequence modulo m, Fibonacci Quart. 6 (1968), 139-141. ·Zbl 0155.36802 |
| [13] | L. Somer, Primes having an incomplete system of residues for a class of second-order recurrences. Applications of Fibonacci Numbers, Vol. 2, A. F. Horadam et al. (eds.), Kluwer Academic Publ., Dordrecht, 1988, 113-141. ·Zbl 0653.10005 |
| [14] | L. Somer, Distribution of residues of certain second-order linear recurrences modulo p – II, Fibonacci Quart. 29 (1991), 72-78. ·Zbl 0728.11010 |
| [15] | L. Somer, Periodicity properties of kth order linear recurrences with irreducible characteristic polynomial over a finite field, Finite Fields, Coding Theory, and Advances in Communications and Computing (Las Vegas, NV, 1991), 195-207, LN in Pure and Appl. Math., 141, Dekker, New York, 1993. ·Zbl 0790.11013 |
| [16] | L. Somer and M. Kˇr´ıˇzek, On moduli for which certain second-order linear recurrences contain a complete system of residues modulo m, Fibonacci Quart. 55 (2017), 209-228. ·Zbl 1401.11049 |
| [17] | W. A. Webb and C. T. Long, Distribution modulo phof the general linear second order recurrence, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 58 (8) (1975), 92-100. ·Zbl 0325.10008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
