A ug 2 00 4 Lucas sequences whose 8 th term is a square

2 6 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...

/ 05 06 01 0 v 3 2 6 A ug 2 00 5 Generalized spacetimes defined by cubic forms and the minimal unitary realizations of their quasiconformal groups

We study the symmetries of generalized spacetimes and corresponding phase spaces defined by Jordan algebras of degree three. The generic Jordan family of formally real Jordan algebras of degree three describe extensions of the Minkowskian spacetimes by an extra “dilatonic” coordinate, whose rotation, Lorentz and conformal groups are SO(d−1), SO(d−1, 1)×SO(1, 1) and SO(d, 2)×SO(2, 1), respective...

On squares in Lucas sequences

Let P and Q be non-zero integers. The Lucas sequence {Un(P,Q)} is defined by U0 = 0, U1 = 1, Un = PUn−1 − QUn−2 (n ≥ 2). The question of when Un(P,Q) can be a perfect square has generated interest in the literature. We show that for n = 2, ..., 7, Un is a square for infinitely many pairs (P,Q) with gcd(P,Q) = 1; further, for n = 8, ..., 12, the only non-degenerate sequences where gcd(P,Q) = 1 a...

ar X iv : n uc l - th / 0 10 80 18 v 1 8 A ug 2 00 1 1 Photon and Electron induced Particle Production on Nuclei ∗

ar X iv : 0 80 8 . 22 17 v 1 [ he p - th ] 1 5 A ug 2 00 8 Scalar QCD 4 on the null - plane

We have studied the null-plane hamiltonian structure of the free Yang-Mills fields and the scalar chromodynamics (SQCD4). Following the Dirac’s procedure for constrained systems we have performed a detailed analysis of the constraint structure of both models and we give the generalized Dirac brackets for the physical variables. In the free Yang-Mills case, using the correspondence principle in ...