Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-26T03:02:59.840Z Has data issue: false hasContentIssue false
  • English
  • Français

Lucas and fibonacci numbers and some diophantine Equations

Published online by Cambridge University Press:  18 May 2009

J. H. E. Cohn
J. H. E. Cohn
Affiliation:
Bedford CollegeLondon, N.W.1
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
Information
Glasgow Mathematical Journal , Volume 7 , Issue 1 , January 1965 , pp. 24 - 28
Copyright
Copyright © Glasgow Mathematical Journal Trust 1965

References

REFERENCES

1.Conn, J. H. E., On square Fibonacci numbers, J. London Math. Soc, 39 (1964), 537–540.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), 161–164.CrossRefGoogle Scholar