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Lucas and fibonacci numbers and some diophantine Equations

Published online by Cambridge University Press:  18 May 2009

J. H. E. Cohn
Affiliation:
Bedford CollegeLondon, N.W.1
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The Lucas numbers υn and the Fibonacci numbers υn are defined by υ1 = 1, υ2= 3, υn+2 = υn+1 + υn and u1 = u2 = 1, un+2 = un+1 + un for all integers n. The elementary properties of these numbers are easily established; see for example [2].However, despite the ease with which many such properties are proved, there are a number of more difficult questions connected with these numbers, of which some are as yet unanswered. Among these there is the well-known conjecture that un is a perfect square only if n = 0, ± 1, 2 or 12. This conjecture was proved correct in [1]. The object of this paper is to prove similar results for υn, ½un and ½υn, and incidentally to simplify considerably the proof for un. Secondly, we shall use these results to solve certain Diophantine equations.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1965

References

1.Conn, J. H. E., On square Fibonacci numbers, J. London Math. Soc, 39 (1964), 537540.CrossRefGoogle Scholar
2.Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers (Oxford, 1954), §10.14.Google Scholar
3.Mordell, L. J., The Diophantine equation y2 = Dx4 + 1, J. London Math. Soc. 39 (1964), 161164.CrossRefGoogle Scholar
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