Geodesic
Lucas and fibonacci numbers and some diophantine Equations
Glasgow mathematical journal, Tome 7 (1965) no. 1, pp. 24-28

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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, 1⁄2un and 1⁄2υn, and incidentally to simplify considerably the proof for un. Secondly, we shall use these results to solve certain Diophantine equations. 
Cohn, J. H. E. Lucas and fibonacci numbers and some diophantine Equations. Glasgow mathematical journal, Tome 7 (1965) no. 1, pp. 24-28. doi: 10.1017/S2040618500035115
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[1] 1.Conn, J. H. E., On square Fibonacci numbers, J. London Math. Soc, 39 (1964), 537–540. Google Scholar | DOI

[2] 2.Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers (Oxford, 1954), §10.14. Google Scholar

[3] 3.Mordell, L. J., The Diophantine equation y2 = Dx4 + 1, J. London Math. Soc. 39 (1964), 161–164. Google Scholar | DOI

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