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Perfect Pell Powers

  • J. H. E. Cohn (a1)
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In the thirty years since it was proved that 0, 1 and 144 were the only perfect squares in the Fibonacci sequence [1, 9], several generalisations have been proved, but many problems remain. Thus it has been shown that 0, 1 and 8 are the only Fibonacci cubes [6] but there seems to be no method available to prove the conjecture that 0, 1, 8 and 144 are the only perfect powers.

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References
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1.Cohn, J. H. E., On Square Fibonacci Numbers, J. London Math. Soc. 39 (1964) 537540.
2.Cohn, J. H. E., Eight Diophantine Equations, Proc. London Math. Soc. (3) 16 (1966) 153166.
3.Cohn, J. H. E., Five Diophantine Equations, Math. Scand. 21 (1967) 6170.
4.Cohn, J. H. E., Squares in some recurrent sequences, Pacific J. Math. 41 (1972) 631646.
5.Ljunggren, W., Zur Theorie de Gleichung x 2 + 1 = Dy 4, Avh. Norske Vid. Akad., Oslo 1, No. 5 (1942).
6.London, Hymie and Finkelstein, Raphael, On Fibonacci and Lucas numbers which are perfect powers, Fibonacci Quart. 5 (1969) 476481.
7.Mordell, L. J., The diophantine equation y 2= Dx 4 + 1, J. London Math. Soc. 39 (1964) 161164.
8.Steiner, Ray and Tzanakis, Nikos, Simplifying the solution of Ljunggren's equation X 2 + 1 = 2Y 4, J. Number Theory, 37 (1991) 123132.
9.Wyler, O., Solution to Problem 5080, Amer. Math. Monthly 71 (1964) 220222.
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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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