@MISC{Mcdanlel93diophantinerepresentation, author = {Wayne L Mcdanlel}, title = {DIOPHANTINE REPRESENTATION OF LUCAS SEQUENCES}, year = {1993} }

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Abstract

Un(P, 0 = PU^iP, 0-QUn_2(P, 0 for n> 2, and the "associated " Lucas sequences {Vn(P, 0} are defined similarly with initial terms equal to 2 and P, for n = 0 and 1, respectively. The sequences of Fibonacci numbers and Lucas numbers are, of course, {Fn} = {U„(l,-1)} and {Ln} = {V„(\,-1)}. Several authors (e.g.,[3], [1], [6]) have discussed the conies whose equations are satisfied by pairs of successive terms of the Lucas sequences. In particular, it has been shown that (x, y)-K, MVH) satisfies y 2-Pxy + Qx 2 +eQ n = 0, where w „ = U„(P, Q)\fe =-1 and w „ = F„(P, 0 if e = P 2-4Q.lt has apparently not been recognized that the hyperbolas y 2- Pxy + Q ^ +eR = 0, where R = 1 if Q = 1 and i? = ±lif(2 =- l characterize the Lucas sequences when ^ =- 1, and the associated Lucas sequences when e- P 2-AQ is square-free; that is, the set of lattice points on these conies is precisely the set of pairs of consecutive terms of {Un(P,±l)} if e-- 1, and of {Vn(P, ± 1)} if e = P 2- 4Q is square-free. Accordingly, we shall prove the converse of the results of [3] and [1] by showing that no lattice points exist for the above hyperbolas if Q = ±1 other than (ww, wn+l) [provided that when wn = V„(P, 0, the discriminant D is square-free]. Using the above results, we then construct, for each of the sequences {Un(P,-1)}, {Un{P, 1)}, and {Vn(P, 1)}, a polynomial in two variables of degree 5, and a polynomial of degree 9 for {Vn(P,-1)} whose positive values, for positive integral values of the variables, are precisely the terms of the sequence. This extends the results of Jones [4] and [5], who obtained a fifthdegree polynomial whose positive values are the Fibonacci numbers and a ninth-degree polynomial whose positive values are the Lucas numbers.