Previous |  Up |  Next

Article

Title: Aritmetické vlastnosti Fibonacciových čísel (Czech)
Title: Arithmetic properties of Fibonacci numbers (English)
Author: Křížek, Michal
Author: Luca, Florian
Author: Somer, Lawrence
Language: Czech
Journal: Pokroky matematiky, fyziky a astronomie
ISSN: 0032-2423
Volume: 50
Issue: 2
Year: 2005
Pages: 127-140
.
Category: math
.
Keyword: Fibonacci number
Keyword: Lucas number
MSC: 11B39
idZBL: Zbl 1265.11024
.
Date available: 2010-12-11T20:59:14Z
Last updated: 2015-11-29
Stable URL: http://hdl.handle.net/10338.dmlcz/141261
.
Reference: [1] André-Jeannin, R.: Irrationalité de la somme des inverses de certaines suites récurrentes.C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), 539–541. MR 0999451
Reference: [2] Badea, C.: The irrationality of certain infinite series.Glasgow Math. J. 29 (1987), 221–228. Zbl 0629.10027, MR 0901668, 10.1017/S0017089500006868
Reference: [3] Brown, J. L.: Zeckendorf’s theorem and some applications.Fibonacci Quart. 2 (1964), 163–168. Zbl 0127.27301
Reference: [4] Brousseau, A.: Tables of Fibonacci entry points (Parts One and Two).The Fibonacci Association 1965.
Reference: [5] Bugeaud, Y., Mignotte, M., Siksek, S.: Sur les nombres de Fibonacci de la forme $q^ky^p$.C. R. Math. Acad. Sci. Paris 339 (2004), 327–330. Zbl 1113.11010, MR 2092057, 10.1016/j.crma.2004.06.007
Reference: [6] Bugeaud, Y., Mignotte, M., Siksek, S.: Classical and modular approaches to exponential diophantine equations, I. Fibonacci and Lucas perfect powers.Submitted to Ann. of Math. (2004).
Reference: [7] Calda, E.: Fibonacciova čísla a Pascalův trojúhelník.Rozhledy mat.-fyz. 71 (1993/94), 15–19.
Reference: [8] Carmichael, R. D.: On the numerical factors of the arithmetic forms ${\alpha ^n\pm \beta ^n}$.Ann. Math. 15 (1913), No. 2, 30–70. MR 1502458
Reference: [9] Drobot, V.: On primes in the Fibonacci sequence.Fibonacci Quart. 38 (2000), 71–72. Zbl 0942.11016, MR 1738649
Reference: [10] Dujella, A.: A proof of the Hoggatt-Bergum conjecture.Proc. Amer. Math. Soc. 127 (1999), 1999–2005. Zbl 0937.11011, MR 1605956, 10.1090/S0002-9939-99-04875-3
Reference: [11] Duverney, D.: Irrationalité de la somme des inverses de la suite de Fibonacci.Elem. Math. 52 (1997), 31–36. Zbl 0886.11041, MR 1438856, 10.1007/s000170050008
Reference: [12] Erdős, P., Graham, R. L.: Old and new problems and results in combinatorial number theory.Monographie 28 de L’Enseign. Math., Imprimerie Kundig, Genéve 1980. MR 0592420
Reference: [13] Good, I. J.: A reciprocal series of Fibonacci numbers.Fibonacci Quart. 12 (1974), 346. Zbl 0292.10009, MR 0351977
Reference: [14] Halton, J. H.: On Fibonacci residues.Fibonacci Quart. 2 (1964), 217–218. Zbl 0119.27905
Reference: [15] Henrici, P.: Discrete variable methods in ordinary differential equations.John Wiley & Sons, New York 1962. Zbl 0112.34901, MR 0135729
Reference: [16] Hogben, L.: An introduction to mathematical genetics.Norton, New York 1946. MR 0019906
Reference: [17] Hoggatt, V. E.: Fibonacci and Lucas numbers.Houghton Mifflin Company, Boston 1969. Zbl 0198.36903
Reference: [18] Jarden, D.: Recurring sequences: a collection of papers.Riveon Lematematika, Jerusalem 1973. MR 0197383
Reference: [19] Jones, J. P.: Diophantine representation of the Fibonacci numbers.Fibonacci Quart. 13 (1975), 84–88. Zbl 0301.10010, MR 0382147
Reference: [20] Jones, J. P., Matiyasevich, Y. V.: Proof of recursive unsolvability of Hilbert’s tenth problem.Amer. Math. Monthly 98 (1991), 689–709. Zbl 0746.03006, MR 1130680, 10.2307/2324421
Reference: [21] Koshy, T.: Fibonacci and Lucas numbers with applications.John Wiley & Sons, Inc., New York 2001. Zbl 0984.11010, MR 1855020
Reference: [22] Křížek, M., Luca, F., Somer, L.: 17 lectures on Fermat numbers.Springer-Verlag, New York 2001. Zbl 1010.11002, MR 1866957
Reference: [23] Křížek, M., Šolcová, A.: Jak spolu souvisí chaos, fraktály a teorie čísel.Sborník semináře Determinismus a chaos, Herbertov 2005, FS ČVUT, Praha 2005, 96–113.
Reference: [24] Lagarias, J. C.: The set of primes dividing the Lucas numbers has density $2/3$.Pacific J. Math. 118 (1985), 449–461. Errata ibid. 162 (1994), 393–396. Zbl 0569.10003, MR 0789184
Reference: [25] Lind, D. A.: The quadratic field ${\mathbb Q}[\sqrt{5}]$ and a certain diophantine equation.Fibonacci Quart. 6 (1968), 86–93. MR 0231784
Reference: [26] Ljunggren, W.: On the diophantine equation ${x^2+4=Ay^2}$.Det. Kgl. Norske Vid.S̄elsk. Forh. 24 (1951), 82–84. MR 0049214
Reference: [27] London, H., Finkelstein, R.: On Fibonacci and Lucas numbers which are perfect powers.Fibonacci Quart. 7 (1969), 476–481, 487. Errata ibid. 8 (1970), 248. Zbl 0206.05402, MR 0255482
Reference: [28] Luca, F.: Proposed problem H-596.Advanced Problem Section, Fibonacci Quart. 41 (2003), 187.
Reference: [29] Luca, F.: Palindromes in Lucas sequences.Monatsh. Math. 138 (2003), 209–223. Zbl 1027.11012, MR 1969517, 10.1007/s00605-002-0490-3
Reference: [30] Luca, F.: Products of factorials in binary recurrence sequences.Rocky Mountain J. Math. 29 (1999), 1387–1411. Zbl 0978.11010, MR 1743376, 10.1216/rmjm/1181070412
Reference: [31] Luo, M.: On triangular Fibonacci numbers.Fibonacci Quart. 27 (1989), 98–108. MR 0995557
Reference: [32] Matiyasevich, Y.: Enumerable sets are diophantine.Soviet Math. Dokl. 11 (1970), 354–358. Zbl 0212.33401
Reference: [33] Matiyasevich, Y.: My collaboration with Julia Robinson.Math. Intell. 14 (1992), no. 4, 38–45. Zbl 0770.01005, MR 1188142, 10.1007/BF03024472
Reference: [34] Matiyasevich, Y. V., Guy, R. K.: A new formula for $\pi $.Amer. Math. Monthly 93 (1986), 631–635. Zbl 0614.10003, MR 1712797
Reference: [35] McDaniel, W.: On Fibonacci and Pell numbers of the form $kx^2$.Fibonacci Quart. 40 (2002), 41–42. Zbl 1068.11010, MR 1885268
Reference: [36] McLaughlin, J.: Small prime powers in the Fibonacci sequence.Preprint, Univ. of Illinois 2002.
Reference: [37] Nemes, I., Pethő, A.: Polynomial values in linear recurrences, II.J. Number Theory 24 (1986), 47–53. MR 0852189
Reference: [38] Pisano, L.: Fibonacci’s Liber abaci.A translation into modern English of Leonardo Pisano’s Book of calculation. Translated by L. E. Sigler, Springer, New York 2002. Zbl 1032.01046, MR 1923794
Reference: [39] Schroeder, M. R.: Number theory in science and communication.Springer Series in Information Sci. 7, second edition, Springer, Berlin 1986. Zbl 0613.10001, MR 0827496
Reference: [40] Stewart, C. L.: On the representation of an integer in two different bases.J. Reine Angew. Math. 319 (1980), 63–72. Zbl 0426.10008, MR 0586115
Reference: [41] Vajda, S.: Fibonacci & Lucas numbers, and the golden section: Theory and applications.John Wiley & Sons, New York 1989. Zbl 0695.10001, MR 1015938
Reference: [42] Vorobiev, N. N.: Fibonacci numbers.Birkhäuser, Basel 2002; Nauka, Moskva 1992. Zbl 1014.11012, MR 1954396
Reference: [43] Wall, D. D.: Fibonacci series modulo $m$.Amer. Math. Monthly 67 (1960), 525–532. Zbl 0101.03201, MR 0120188, 10.2307/2309169
Reference: [44] Williams, H. C.: A note on the Fibonacci quotients $F_{p-\epsilon }/p$.Canad. Math. Bull. 25 (1982), 366–370. MR 0668957, 10.4153/CMB-1982-053-0
Reference: [45] Zhu, Z., Cao, L., Liu, X., Zhu, W.: Topological invariance of the Fibonacci sequences of the periodic buds in general Mandelbrot sets.J. Northeast Univ. Na. Sci. 22 (2001), 497–500. MR 1869910
.

Files

Files Size Format View
PokrokyMFA_50-2005-2_3.pdf 390.9Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo