FFF. Fibonacci: di Fiore in Fiore

Ribenboim, Paulo

Bollettino dell'Unione Matematica Italiana, Tome 5-A (2002), p. 329-353 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract

In occasione della commemorazione dell’800-esimo anniversario della pubblicazione del Liber Abaci, desidero richiamare l’attenzione del lettore su alcuni dei fatti che preferisco riguardanti numeri di Fibonacci. Tali fatti includono la presenza di quadrati, di multipli di quadrati e di numeri potenti tra i numeri di Fibonacci, la rappresentazione di numeri reali e la costruzione di numeri trascendenti mediante numeri di Fibonacci, la possibilità di costruire una serie zeta ed un dominio a fattorizzazione unica associati ai numeri di Fibonacci

Publié le : 2002-08-01

MR 1930669 | Zbl 1194.01008

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     author = {Paulo Ribenboim},
     title = {FFF. Fibonacci: di Fiore in Fiore},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {5-A},
     year = {2002},
     pages = {329-353},
     zbl = {1194.01008},
     mrnumber = {1930669},
     language = {it},
     url = {http://dml.mathdoc.fr/item/BUMI_2002_8_5A_2_329_0}
}

Ribenboim, Paulo. FFF. Fibonacci: di Fiore in Fiore. Bollettino dell'Unione Matematica Italiana, Tome 5-A (2002) pp. 329-353. http://gdmltest.u-ga.fr/item/BUMI_2002_8_5A_2_329_0/

Bibliographie

[Ho69] Hoggatt, V. E., Fibonacci and Lucas Numbers, Houghton-Mifflin, Boston (1969). | Zbl 0198.36903

[Vo63] Vorob’Ev, N. N., The Fibonacci numbers, D. C. Heath, Boston (1963). | MR 168515 | Zbl 0826.11006

[AJ91] Andre-Jeannin, R., A note on the irrationality of certain Lucas infinite series, Fibonacci Q., 29 (1991), 132-136. | MR 1119398 | Zbl 0725.11034

[BMS88] Brillhart, J. - Montgomery, P. L. - Silverman, R. D., Tables of Fibonacci and Lucas factorizations, Math. Comp., 50 (1983), 251-260. | MR 917832 | Zbl 0634.10002

[Ca13] Carmichael, R. D., On the numerical factors of arithmetic forms $α^{n} \pm β^{n}$ , Annals of Math. (2), 15 (1913), 30-70. | JFM 44.0216.01 | MR 1502458

[CE63] Cashwell, E. D. - Everett, C. J., Formal power series, Pacific J. Math., 13 (1963), 45-69. | MR 155852 | Zbl 0117.02603

[Co64a] Cohn, J. H. E., On square Fibonacci numbers, J. London Math. Soc., 39 (1964), 537-540. | MR 163867 | Zbl 0127.26705

[Co64b] Cohn, J. H. E., Square Fibonacci numbers, etc., Fibonacci Q., 2 (1964), 109-113. | MR 161819 | Zbl 0126.07201

[Co65] Cohn, J. H. E., Lucas and Fibonacci numbers and some Diophantine equations, Proc. Glasgow Math. Assoc., 7 (1965), 24-28. | MR 177944 | Zbl 0127.01902

[El91] Elkies, N. D., (ABC) implies Mordell, Intern. Math. Res. Notices, 7 (1991), 99-109. | MR 1141316 | Zbl 0763.11016

[Ha66] Halton, J. H., On the divisibility properties of Fibonacci numbers, Fibonacci Quart., 4 (1966), 217-240. | MR 201373 | Zbl 0147.02202

[Kn64] Knuth, D., Transcendental numbers based on the Fibonacci numbers, Fibonacci Q., 2 (1964), 43-44. | Zbl 0121.05103

[LW81] Lagarias, J. C. - Weissel, D. P., Fibonacci and Lucas cubes, Fibonacci Q., 19 (1981), 39-43. | MR 606108 | Zbl 0445.10015

[LF69] London, H. - Finkelstein, R., On Fibonacci and Lucas numbers which are perfect powers, Fibonacci Q., 7 (1969), 476-481 e 487. | MR 255482 | Zbl 0206.05402

[Lu78] Lucas, E., Théorie des fonctions numériques simplement périodiques, American J. Math., 1 (1878), 184-240, 289-321. | JFM 10.0134.05 | MR 1505176

[Ma85] Masser, D. W., Open problems, Proc. Symp. Anal. Number Theory (W. W. W. L. Chern editor), Imperial College, London (1985).

[McR98] Mc Daniel, W. L. - Ribenboim, P., Square classes in Lucas sequences, J. Number Theory, 73 (1998), 14-27. | MR 1654862 | Zbl 0923.11037

[Mo96] Mollin, R. A., Masser’s conjecture used to prove results about powerful numbers, J. Math. Sci., 7 (1996), 29-32. | MR 1433216

[Oe88] Oesterle, J., Nouvelles approches du «théorème» de Fermat, Sém. Bourbaki 694, Astérisque, 161-162 (1988), 165-186. | MR 992208 | Zbl 0668.10024

[Pe82] Petho, A., Perfect powers in second order linear recurrences, J. Number Theory, 15 (1982), 5-13. | MR 666345 | Zbl 0488.10009

[Pe83] Petho, A., Full cubes in the Fibonacci sequence, Publ. Math. Debrecen, 30 (1983), 117-127. | MR 733078 | Zbl 0549.10007

[Ri89] Ribenboim, P., Representations of real numbers by means of Fibonacci numbers, Enseign. Math., 31 (1989), 249-259. | MR 819353 | Zbl 0587.10003

[Ri95] Ribenboim, P., The Fibonacci numbers and the Arctic Ocean, Symp. Gaussiana, Conf. A (M. Behara, R. Fritsch and R. G. Lintz editors) (1995), 41-83. | MR 1352481 | Zbl 0858.11011

[Ri96] Ribenboim, P., The New Book of Prime Number Records, Springer Verlag, New York (1996). | MR 1377060 | Zbl 0856.11001

[Ri99] Ribenboim, P., An algorithm to determine the points with integral coordinates in certain elliptic curves, J. Number Theory, 74 (1999), 19-39. | MR 1670552 | Zbl 0923.11036

[Ri00] Ribenboim, P., My Numbers, my Friends, Springer-Verlag, New York (2000). | MR 1761897 | Zbl 0947.11001

[Ri01a] Ribenboim, P., ABC candies, J. Number Theory, 81 (2001), 47-60. | MR 1743505 | Zbl 1007.11013

[Ri01b] Ribenboim, P., The (ABC) conjecture and the radical index of integers, Acta Arith., 96 (2001), 389-404. | MR 1811881 | Zbl 1031.11056

[RMc96] Ribenboim, P. - Mc Daniel, W. L., The square terms in Lucas sequences, J. Number Theory, 58 (1996), 104-123. | MR 1387729 | Zbl 0851.11011

[RW99] Ribenboim, P. - Walsh, P. G., The ABC conjecture and the powerful part of terms in binary recurring sequences, J. Number Theory, 74 (1999), 134-147. | MR 1670556 | Zbl 0923.11033

[SS83] Shorey, T. N. - Stewart, C. L., On the Diophantine equation $a x^{2} t + b x^{} t y + c y^{2} = 1$ and pure powers in recurrence sequences, Math. Scand., 52 (1983), 24-36. | MR 697495 | Zbl 0491.10016

[Wy64] Wyler, O., Squares in the Fibonacci series, Amer. Math. Monthly, 7 (1964), 220-222.