2 6 A ug 2 00 4 Lucas sequences whose 8 th term is a square
نویسندگان
چکیده
For each positive integer n ≤ 7 we describe all Lucas sequences with (P,Q) = 1 having the property that Un(P,Q) is a perfect square. The arguments are elementary. We also find all Lucas sequences such that U8(P,Q) is a perfect square. This reduces to a number of problems of similar type, namely, finding all points on an elliptic curve defined over a quartic number field subject to a “Q-rationality” condition on the X-coordinate. This is achieved by p-adic computations (for a suitable prime p) using the formal group of the elliptic curve.
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A ug 2 00 4 Lucas sequences whose 8 th term is a square
For each positive integer n ≤ 7 we describe all Lucas sequences with (P,Q) = 1 having the property that Un(P,Q) is a perfect square. The arguments are elementary. We also find all Lucas sequences such that U8(P,Q) is a perfect square. This reduces to a number of problems of similar type, namely, finding all points on an elliptic curve defined over a quartic number field subject to a “Q-rational...
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The sequence {Un(1,−1)} is the familiar Fibonacci sequence, and it was proved by Cohn [12] in 1964 that the only perfect square greater than 1 in this sequence is U12 = 144. The question arises, for which parameters P , Q, can Un(P,Q) be a perfect square? This has been studied by several authors: see for example Cohn [13] [14] [15], Ljunggren [22], and Robbins [25]. Using Baker’s method on line...
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