On squares in Lucas sequences
by
Andrew Bremner
Arizona State University
Coauthors: Nikos Tzanakis

Let P and Q be non-zero integers. The Lucas sequence {U_n(P, Q)} is defined by

\labelLucasU0=0,     U1=1,     Un = P Un-1-Q Un-2     (n >= 2).

(\theequation)

The sequence {U_n(1, -1)} is the familiar Fibonacci sequence, and it was proved by Cohn 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 U_n(P, Q) be a perfect square? Ribenboim and McDaniel using only elementary methods show that when P and Q are odd, and P²-4Q > 0, then U_n can be square only for n=0, 1, 2, 3, 6, or 12. They characterize fully the instances when U_n=square for n=2, 3, 6.
There seems little mention in the literature of when under general hypotheses U_n(P, Q) can be a perfect square. A small computer search reveals sequences with U_n(P, Q) a perfect square, (P, Q)=1, only for n=0, ... , 8, and n=12. We present here arguments that for n <= 9, n=12 find all instances of U_n(P, Q)=square, subject only to the restriction (P, Q)=1. The cases n <= 7 can be treated entirely elementarily. The cases n=8, 9, 12 reduce to finding all rational points on several curves of genus 2. Methods first introduced by Bruin and Flynn are used, demanding working with the formal group of elliptic curves over number fields.

Paper reference: arXiv:math.NT/0408371, arXiv:math.NT/0405306

Date received: June 9, 2004