Powers in recurrence sequences: Pell equations
نویسندگان
چکیده
منابع مشابه
Powers in Recurrence Sequences: Pell Equations
In this paper, we present a new technique for determining all perfect powers in so-called Pell sequences. To be precise, given a positive nonsquare integer D, we show how to (practically) solve Diophantine equations of the form x −Dy = 1 in integers x, y and n ≥ 2. Our method relies upon Frey curves and corresponding Galois representations and eschews lower bounds for linear forms in logarithms...
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Let {uk} be a Lucas sequence. A standard technique for determining the perfect powers in the sequence {uk} combines bounds coming from linear forms in logarithms with local information obtained via Frey curves and modularity. The key to this approach is the fact that the equation uk = xn can be translated into a ternary equation of the form ay2 = bx2n + c (with a, b, c ∈ Z) for which Frey curve...
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We show how the Gelfond-Baker theory and diophantine approximation techniques can be applied to solve explicitly the diophantine equation G, = wp" ... p', (where (G,, }I='o is a binary recurrence sequence with positive discriminant), for arbitrary values of the parameters. We apply this to the equation x2 + k = ... ps', which is a generalization of the Ramanujan-Nagell equation x2 + 7 = 2Z. We ...
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Let R and S be positive integers with R < S. We shall call the simultaneous Diophantine equations x −Ry = 1, z − Sy = 1 simultaneous Pell equations in R and S. Each such pair has the trivial solution (1, 0, 1) but some pairs have nontrivial solutions too. For example, if R = 11 and S = 56, then (199, 60, 449) is a solution. Using theorems due to Baker, Davenport, and Waldschmidt, it is possible...
متن کاملSolving constrained Pell equations
Consider the system of Diophantine equations x2 − ay2 = b, P (x, y) = z2, where P is a given integer polynomial. Historically, such systems have been analyzed by using Baker’s method to produce an upper bound on the integer solutions. We present a general elementary approach, based on an idea of Cohn and the theory of the Pell equation, that solves many such systems. We apply the approach to th...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 2004
ISSN: 0002-9947,1088-6850
DOI: 10.1090/s0002-9947-04-03586-x