The Fibonacci Numbers and the Arctic Ocean

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  1. 1202 Leonardo Pisano (Fibonacci). Liber Abbaci (21228). Tipografia delle Scienze Matematiche e Fisiche, Rome, 1857 edition. B. Boncompagni, editor.Google Scholar
  2. 1657 Frénicle de Bessy. Solutio duorum problematum circa numeros cubos et quadratos. Bibliothèque Nationale de Paris.Google Scholar
  3. 1843 J. P. M. Binet. Mémoire sur ľintrégation des équations linéaires aux différences finies, ďun ordre quelconque, á coefficients variables. C. R. Acad. Sci. Paris, 17:559–567.Google Scholar
  4. 1844 E. Catalan. Note extraite ďune lettre addressée á ľéditeur. J. reine u. angew. Math., 27:192.MATHGoogle Scholar
  5. 1870 G. C. Gérono. Note sur la résolution en nombres entiers et positifs de ľéquation x m=y n+1. Nouv. Ann. de Math. (2), 9:469–471, and 10:204–206 (1871).Google Scholar
  6. 1878 E. Lucas. Théorie des fonctions numériques simplement périodiques. Amer. J. of Math., 1:184–240 and 289–321.MATHMathSciNetGoogle Scholar
  7. 1886 A. S. Bang. Taltheoretiske Untersogelser. Tidskrift Math., Ser. 5, 4:70–80 and 130–137.MATHGoogle Scholar
  8. 1892 K. Zsigmondy. Zur Theorie der Potenzreste. Monatsh. f. Math., 3:265–284.CrossRefMATHMathSciNetGoogle Scholar
  9. 1904 G. D. Birkho and H. S. Vandiver. On the integral divisors of a n-b n. Ann. Math. (2), 5:173–180.CrossRefGoogle Scholar
  10. 1909 A. Wieferich. Zum letzten Fermatschen Theorem. J. reine u. angew. Math., 136:293–302.MATHGoogle Scholar
  11. 1913 R. D. Carmichael. On the numerical factors of arithmetic forms αn±±βn. Ann. of Math. (2), 15:30–70.CrossRefMATHMathSciNetGoogle Scholar
  12. 1920 T. Nagell. Note sur ľéquation indéterminée \( \frac{{x^n - 1}} {{x - 1}} = y^q \). Norsk Mat. Tidsskr., 2:75–78.Google Scholar
  13. 1921a T. Nagell. Des équations indéterminées x 2+x+1=y n et x 2+x+1=3y n. Norsk Mat. Forenings Skrifter, Ser. I, 1921, No. 2, 14 pages.Google Scholar
  14. 1921b T. Nagell. Sur ľéquation indéterminée \( \frac{{x^n - 1}} {{x - 1}} = y^2 \). Norsk Mat. Forenings Skrifter, Ser. I, 1921, No. 3, 17 pages.Google Scholar
  15. 1930 D. H Lehmer. An extended theory of Lucas’ functions. Ann. of Math., 31:419–448.CrossRefMATHMathSciNetGoogle Scholar
  16. 1935 K. Mahler. Eine arithmetische Eigenschaft der Taylor-Koeffizienten rationaler Funktionen. Nederl. Akad. Wetensch. Amsterdam Proc., 38:50–60.MATHGoogle Scholar
  17. 1938 G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers. Clarendon Press, Oxford, 5th (1979) edition.Google Scholar
  18. 1942a W. Ljunggren. Einige Bemerkungen über die Darstellung ganzer Zahlen durch binäre kubische Formen mit positiver Diskriminante. Acta Math., 75:1–21.MATHMathSciNetGoogle Scholar
  19. 1942b W. Ljunggren. Über die Gleichung x 4Dy 2=1. Arch. Math. Naturvid., 45(5):61–70.MATHMathSciNetGoogle Scholar
  20. 1942c W. Ljunggren. Zur Theorie der Gleichung x 2+1=Dy 4. Avh. Norsk Vid. Akad. Oslo., 1(5):1–27.MathSciNetGoogle Scholar
  21. 1943 W. Ljunggren. New propositions about the indeterminate equationxn x n −1/x−1=y q. Norsk Mat. Tidsskr., 25:17–20.MATHMathSciNetGoogle Scholar
  22. 1950 H.-J. Kanold. Sätze über Kreisteilungspolynome und ihre Anwendungen auf einige zahlentheoretische Probleme. J. reine u. angew. Math., 187:355–366.MathSciNetGoogle Scholar
  23. 1953 C. G. Lekkerkerker. Prime factors of elements of certain sequences of integers. Nederl. Akad. Wetensch. Proc. (A), 56:265–280.MATHMathSciNetGoogle Scholar
  24. 1954 M. Ward. Prime divisors of second order recurring sequences. Duke Math. J., 21:607–614.CrossRefMATHMathSciNetGoogle Scholar
  25. 1955 E. Artin. The order of the linear group. Comm. Pure Appl. Math., 8:335–365.Google Scholar
  26. 1955 M. Ward. The intrinsic divisors of Lehmer numbers. Ann. of Math. (2), 62:230–236.CrossRefMATHMathSciNetGoogle Scholar
  27. 1958 D. Jarden. Recurring Sequences. Riveon Lematematike, Jerusalem. 3 1973, revised and enlarged by J. Brillhart, Fibonacci Assoc., San Jose, CA.Google Scholar
  28. 1960 A. A. Brauer. Note on a number theoretical paper of Sierpiʼnski. Proc. Amer. Math. Soc., 11:406–409.CrossRefMATHMathSciNetGoogle Scholar
  29. 1960 Chao Ko. On the Diophantine equation x 2=y n+1. Acta Sci. Natur. Univ. Szechuan, 2:57–64.Google Scholar
  30. 1961 L. K. Durst. Exceptional real Lucas sequences. Pacific J. Math., 11:489–494.MATHMathSciNetGoogle Scholar
  31. 1961 M. Ward. The prime divisors of Fibonacci numbers. Pacific J. Math., 11:379–389.MATHMathSciNetGoogle Scholar
  32. 1962 A. Rotkiewicz. On Lucas numbers with two intrinsic prime divisors. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys., 10:229–232.MATHMathSciNetGoogle Scholar
  33. 1962a A. Schinzel. The intrinsic divisions of Lehmer numbers in the case of negative discriminant. Ark. Math., 4:413–416.MATHMathSciNetGoogle Scholar
  34. 1962b A. Schinzel. On primitive prime factors of a nb n. Proc. Cambridge Phil. Soc., 58:555–562.MATHMathSciNetCrossRefGoogle Scholar
  35. 1963a A. Schinzel. On primitive prime factors of Lehmer numbers, I. Acta Arith., 8:213–223.MATHMathSciNetGoogle Scholar
  36. 1963b A. Schinzel. On primitive prime factors of Lehmer numbers, II. Acta Arith., 8:251–257.MATHMathSciNetGoogle Scholar
  37. 1963 N. N. Vorob’ev. The Fibonacci Numbers. D. C. Heath, Boston.Google Scholar
  38. 1964a J. H. E. Cohn. On square Fibonacci numbers. J. London Math.Soc., 39:537–540.MATHMathSciNetGoogle Scholar
  39. 1964b J. H. E. Cohn. Square Fibonacci numbers etc. Fibonacci Q., 2:109–113.MATHGoogle Scholar
  40. 1964 Chao Ko. On the Diophantine equation x 2=y n+1. Scientia Sinica (Notes), 14:457–460.Google Scholar
  41. 1964 O. Wyler. Squares in the Fibonacci series. Amer. Math. Monthly, 7:220–222.MathSciNetGoogle Scholar
  42. 1965 J. H. E. Cohn. Lucas and Fibonacci numbers and some Diophantine equations. Proc. Glasgow Math. Assoc., 7:24–28.MATHMathSciNetCrossRefGoogle Scholar
  43. 1965 P. Erdös. Some recent advances and current problems in number theory. In Lectures on Modern Mathematics, Vol. III, edited by T. L. Saaty, 169–244. Wiley, New York.Google Scholar
  44. 1965 H. Hasse. Über die Dichte der Primzahlen p, für die eine vorgegebene ganzrationale Zahl a≠0 von durch eine vorgegebene Primzahl l≠2 teilbarer bzw. unteilbarer Ordnung mod p ist. Math. Annalen, 162:74–76.CrossRefMATHMathSciNetGoogle Scholar
  45. 1966 J. H. E. Cohn. Eight Diophantine equations. Proc. London Math. Soc. (3), 16:153–166, and 17:381.MATHMathSciNetGoogle Scholar
  46. 1966 H. Hasse. Über die Dichte der Primzahlen p, för die eine vorgegebene ganzrationale Zahl a≠0 von gerader bzw. ungerader Ordnung mod p ist. Math. Annalen, 168:19–23.CrossRefMathSciNetGoogle Scholar
  47. 1966 K. Mahler. A remark on recursive sequences. J. Math. Sci., 1:12–17.MATHMathSciNetGoogle Scholar
  48. 1966 E. Selmer. Linear Recurrences over Finite Fields. Lectures Notes, Department of Mathematics, University of Bergen.Google Scholar
  49. 1967 J. H. E. Cohn. Five Diophantine equations. Math. Scand., 21:61–70.MATHMathSciNetGoogle Scholar
  50. 1967 C. Hooley. On Artin’s conjecture. J. reine u. angew. Math., 225:209–220.MATHMathSciNetGoogle Scholar
  51. 1968 J. H. E. Cohn. Some quartic Diophantine equations. Pacific J. Math., 26:233–243.MATHMathSciNetGoogle Scholar
  52. 1968 L. P. Postnikova and A. Schinzel. Primitive divisors of the expression a nb n. Math. USSR-Sb., 4:153–159.CrossRefMATHGoogle Scholar
  53. 1968 A. Schinzel. On primitive prime factors of Lehmer numbers, III. Acta Arith., 15:49–70.MATHMathSciNetGoogle Scholar
  54. 1969 V. E. Hoggatt. Fibonacci and Lucas Numbers. Houghton-Mifflin, Boston.MATHGoogle Scholar
  55. 1969 R. R. Laxton. On groups of linearrecurrences, I. Duke Math. J., 36:721–736.CrossRefMATHMathSciNetGoogle Scholar
  56. 1969 H. London and R. Finkelstein (alias R.Steiner). On Fibonacci and Lucas numbers whichare perfect powers. Fibonacci Q., 7:476–481 and 487.Google Scholar
  57. 1972 J. H. E. Cohn. Squares in some recurrence sequences. Pacific J. Math., 41:631–646.MATHMathSciNetGoogle Scholar
  58. 1972 K. Inkeri. On the Diophantic equation \( a\frac{{x^n - 1}} {{x - 1}} = y^m \). Acta Arith., 21:299–311.MATHMathSciNetGoogle Scholar
  59. 1973 A. Baker. A sharpening for the bounds of linear forms in logarithms, II. Acta Arith., 24:33–36.MATHMathSciNetGoogle Scholar
  60. 1973 H. London and R. Finkelstein (alias R. Steiner). Mordell’s Equation y 2k=x 3. Bowling Green State University Press, Bowling Green, OH.Google Scholar
  61. 1974 A. Schinzel. Primitive divisions of the expression A n −B n in algebraic number fields. J. reine u. angew. Math., 268/269: 27–33.MathSciNetGoogle Scholar
  62. 1975 A. Baker. Transcendental Number Theory. Cambridge Univ. Press, Cambridge.MATHGoogle Scholar
  63. 1975 C. L. Stewart. The greatest prime factor of a n −b n. Acta Arith., 26:427–433.MATHGoogle Scholar
  64. 1976 E. Z. Chein. A note on the equation x 2=y n+1. Proc. Amer. Math. Soc., 56:83–84.CrossRefMATHMathSciNetGoogle Scholar
  65. 1976 S. V. Kotov. Über die maximale Norm der Idealteiler des Polynoms αx my n mit den algebraischen Koeffizenten. Acta Arith., 31:210–230.MathSciNetGoogle Scholar
  66. 1976 M. Langevin. Quelques applications des nouveaux résultats de van der Poorten. Sém. Delange-Pisot-Poitou, 17 e année, 1976, No. G12, 1–11.Google Scholar
  67. 1976 P. J. Stephens. Prime divisors of second order linear recurrences, I. and II. J. Nb. Th., 8:313–332 and 333–345.CrossRefMATHMathSciNetGoogle Scholar
  68. 1976 R. Tijdeman. On the equation of Catalan. Acta Arith., 29: 197–209.MathSciNetGoogle Scholar
  69. 1977 A. Baker. The theory of linear forms in logarithms. In Transcendence Theory: Advances and Applications (Proceedings of a conference held in Cambridge 1976), edited by A. Baker and D. W. Masser, 1–27. Academic Press, New York.Google Scholar
  70. 1977 T. N. Shorey, A. J. van der Porten, R. Tijdeman, and A. Schinzel. Applications of the Geľfond-Baker method to Diophantine equations. In Transcendence theory: Advances and Applications, edited by A. Baker and D. W. Masser, 59–77. Academic Press, New York.Google Scholar
  71. 1977a C. L. Stewart. On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers. Proc. London Math. Soc., 35:425–447.MATHMathSciNetGoogle Scholar
  72. 1977b C. L. Stewart. Primitive divisors of Lucas and Lehmer numbers. In Trancendence Theory: Advances and Applications, edited by A. Baker and D. W. Masser, 79–92. Academic Press, New York.Google Scholar
  73. 1977 A. J. van der Poorten. Linear forms in logarithms in p-adic case. In Transcendence Theory: Advances and Applications, edited by A. Baker and D. W. Masser, 29–57. Academic Press, New York.Google Scholar
  74. 1978 P. Kiss and B. M. Phong. On a function concerning second order recurrences. Ann. Univ. Sci. Budapest. Eötvös Sect Math., 21:119–122.MATHGoogle Scholar
  75. 1978 N. Robbins. On Fibonacci numbers which are powers. Fibonacci Q., 16:515–517.MATHMathSciNetGoogle Scholar
  76. 1980 R. Steiner. On Fibonacci numbers of the form v 2+1. In A Collection of Manuscripts Related to the Fibonacci Sequence, edited by W. E. Hogatt and M. Bicknell-Johnson, 208–210. The Fibonacci Association, Santa Clara, CA.Google Scholar
  77. 1980 C. L. Stewart. On some Diophantine equations and related recurrence sequences. In Séminaire de Théorie des Nombres Paris 1980/81 (Séminare Delange-Pisot-Poitou), Progress in Math., 22:317–321 (1982). Birkhäuser, Boston.Google Scholar
  78. 1980 M. Waldschmidt. A lower bound for linear forms in logarithms. Acta Arith., 37:257–283.MATHMathSciNetGoogle Scholar
  79. 1981 K. Gyæry, P. Kiss, and A. Schinzel. On Lucas and Lehmer sequences and their applications to Diophantine equations. Cplloq. Math., 45:75–80.Google Scholar
  80. 1981 J. C. Lagarias and D. P. Weissel. Fibonacci and Lucas cubes. Fibonacci Q., 19:39–43.MATHGoogle Scholar
  81. 1981 H. Lüuneburg. Ein einfacher Beweis füur den Satz von Zsigdmondy über primitive Primteiler von A n −B n. In Ge-ometries and Groups, Lect. Notes in Math., 893:219–222, edited by M. Aigner and D. Jungnickel. Springer-Verlag, New York.Google Scholar
  82. 1981 T. N. Shorey and C. L. Stewart. On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers, II. J. London Math. Soc., 23:17–23.MathSciNetMATHGoogle Scholar
  83. 1982 K. Györy. On some arithmetical properties of Lucas and Lehmer numbers. Acta Arith., 40:369–373.MATHGoogle Scholar
  84. 1982 A. Pethö. Perfect powers in second order linear recurrences. J. Nb. Th., 15:5–13.CrossRefMATHGoogle Scholar
  85. 1982 C. L. Stewart. On divisors of terms of linear recurrence sequences. J. reine u. angew. Math., 333:12–31.MATHGoogle Scholar
  86. 1983 A. Pethö. Full cubes in the Fibonacci sequence. Publ. Math. Debrecen, 30:117–127.MATHMathSciNetGoogle Scholar
  87. 1983a N. Robbins. On Fibonacci numbers of the form px 2, where p is a prime. Fibonacci Q., 21:266–271.MATHMathSciNetGoogle Scholar
  88. 1983b N. Robbins. On Fibonacci numbers which are powers, II. Fibonacci Q., 21:215–218.MATHMathSciNetGoogle Scholar
  89. 1983 A. Rotkiewicz. Applications of Jacobi symbol to Lehmer’s numbers. Acta Arith., 42:163–187.MATHMathSciNetGoogle Scholar
  90. 1983 T. N. Shorey and C. L. Stewart. On the Diophantine equation ax 2t+bx t y+cy 2=1 and pure powers in recurrence sequences. Math. Scand., 52:24–36.MathSciNetMATHGoogle Scholar
  91. 1984 N. Robbins. On Pell numbers of the form px 2, where p is prime. Fibonacci Q. (4), 22:340–348.MATHMathSciNetGoogle Scholar
  92. 1985 J. C. Lagarias. The set of primes dividing the Lucas numbers has density 2/3. Pacific J. Math., 118:19–23.MathSciNetGoogle Scholar
  93. 1985 D. W. Masser. Open problems. In Proceedings Symposium Analytic Number Theory, edited by W. W. L. Chen, London. Imperial College.Google Scholar
  94. 1985 C. L. Stewart. On the greatest prime factor of terms of a linear recurrence sequence. Rocky Mountain J. Math., 15:599–608.MATHMathSciNetCrossRefGoogle Scholar
  95. 1986 T. N. Shorey and R. Tijdeman. Exponential Diophantine Equations. Cambridge University Press, Cambridge.MATHGoogle Scholar
  96. 1986 C. L. Stewart and R. Tijdeman. On the Oesterlé-Masser conjecture. Monatshefte Math., 102:251–257.CrossRefMathSciNetMATHGoogle Scholar
  97. 1987 A. Rotkiewicz. Note on the Diophantine equation 1+x+x 2+⋯+x m=y m. Elem. of Math., 42:76.MATHMathSciNetGoogle Scholar
  98. 1988 J. Brillhart, P. L. Montgomery, and R. D. Silverman. Tables of Fibonacci and Lucas factorizations. Math. of Comp., 50:251–260.CrossRefMathSciNetGoogle Scholar
  99. 1988 M. Goldman. Lucas numbers of the form px 2, where p=3, 7, 47 or 2207. C. R. Math. Rep. Acad. Sci. Canada, 10:139–141.MATHMathSciNetGoogle Scholar
  100. 1988 J. Oesterlé. Nouvelles approches du “théorème” de Fermat. Séminaire Bourbaki, 40ème anée, 1987/8, No. 694, Astérisque, 161–162, 165–186.MATHGoogle Scholar
  101. 1989a P. Ribenboim. Square-classes of Fibonacci numbers and Lucas numbers. Portug. Math., 46:159–175.MATHMathSciNetGoogle Scholar
  102. 1989b P. Ribenboim. Square-classes of a n−1/a−1 and a n+1. J. Sichuan Univ. Nat. Sci. Ed., 26:196–199. Spec. Issue.MATHMathSciNetGoogle Scholar
  103. 1989 N. Tzanakis and B. M. M. de Weger. On the practical solution of the Thue equation. J. Nb. Th., 31:99–132.CrossRefMATHGoogle Scholar
  104. 1991 W. D. Elkies. ABC implies Mordell. Internat. Math. Res. Notices (Duke Math. J.), 7:99–109.CrossRefMATHMathSciNetGoogle Scholar
  105. 1991 A. Pethö. The Pell sequence contains only trivial perfect powers. In Colloquia on Sets, Graphs and Numbers, Soc. Math., János Bolyai, 561–568. North-Holland, Amsterdam.Google Scholar
  106. 1991a P. Ribenboim. The Little Book of Big Primes. Springer-Verlag, New York.MATHGoogle Scholar
  107. 1991b P. Ribenboim and W. L McDaniel. Square-classes of Lucas sequences. Portug. Math., 48:469–473.MathSciNetMATHGoogle Scholar
  108. 1992 R. André-Jeannin. On the equations U n=U q x 2, where q is odd and V n=V q x 2, where q is even. Fibonacci Q., 30:133–135.MATHGoogle Scholar
  109. 1992 W. L. McDaniel and P. Ribenboim. Squares and double squares in Lucas sequences. C. R. Math. Rep. Acad. Sci. Canada, 14:104–108.MathSciNetMATHGoogle Scholar
  110. 1994 P. Ribenboim. Catalan’s Conjecture. Academic Press, Boston.MATHGoogle Scholar
  111. 1995 P. M. Voutier. Primitive divisors of Lucas and Lehmer sequences. Math. of Comp., 64:869–888.CrossRefMATHMathSciNetGoogle Scholar
  112. 1998a W. L. McDaniel and P. Ribenboim. Square classes in Lucas sequences having odd parameters. J. Nb. Th., 73:14–23.CrossRefMathSciNetMATHGoogle Scholar
  113. 1998b W. L. McDaniel and P. Ribenboim. Squares in Lucas sequences having one even parameter. Colloq. Math., 78:29–34.MathSciNetMATHGoogle Scholar
  114. 1999 Y. Bugeaud and M. Mignotte. On integers with identical digits. Preprint.Google Scholar
  115. 1999 H. Dubner and W. Keller. New Fibonacci and Lucas primes. Math. of Comp., 68:417–427.CrossRefMathSciNetMATHGoogle Scholar
  116. 1999a P. Ribenboim. Números primos, Mistérios e Récordes. Instituto de Matématica Puru e Aplicado, Rio de Janeiro.Google Scholar
  117. 1999b P. Ribenboim and P. G. Walsh. The ABC conjecture and the powerful part of terms in binary recurring sequences. J. Nb. Th., 74:134–147.CrossRefMathSciNetMATHGoogle Scholar

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