منابع مشابه
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...
متن کاملSquares in Lucas Sequences with Rational Roots
In 1997, Darmon and Merel proved the stunning result that the Diophantine equation x + y = z has no nontrivial integer solutions for n ≥ 4. This can be interpreted as saying that if {vn} represents a Lucas sequence of the second kind, defined by a quadratic polynomial with rational roots, then the equation vn = x , with x an integer, implies that n ≤ 3. The goal of the present paper is to prove...
متن کاملOn generalized Lucas sequences
We introduce the notions of unsigned and signed generalized Lucas sequences and prove certain polynomial recurrence relations on their characteristic polynomials. We also characterize when these characteristic polynomials are irreducible polynomials over a finite field. Moreover, we obtain the explicit expressions of the remainders of Dickson polynomials of the first kind divided by the charact...
متن کاملOn Lucas Sequences Computation
This paper introduces an improvement to a currently published algorithm to compute both Lucas “sister” sequences Vk and Uk. The proposed algorithm uses Lucas sequence properties to improve the running time by about 20% over the algorithm published in [1].
متن کاملOn integral points on biquadratic curves and near multiples of squares in Lucas sequences
We describe an algorithmic reduction of the search for integral points on a curve y2 = ax4 + bx2 + c with ac(b2 − 4ac) 6= 0 to solving a finite number of Thue equations. While existence of such reduction is anticipated from arguments of algebraic number theory, our algorithm is elementary and to best of our knowledge is the first published algorithm of this kind. In combination with other metho...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2007
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2006.10.007