منابع مشابه
Simultaneous Pell equations
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...
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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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IF Introduction sn IWWW fremner I onsidered rithmeti progressions on ellipti urvesF fremner onstruted ellipti urves with rithmeti progressions of length UD iFeF rtionl points @X; Y A whose XE oordintes re in rithmeti progressionF sn following pper fremnerD ilvermn nd znkis P showed tht sugroup of the ellipti urve E@QA with E X Y 2 a X@X 2 n 2 A of rnk I doe...
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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...
متن کاملOn the resolution of simultaneous Pell equations ∗
We descibe an alternative procedure for solving automatically simultaneous Pell equations with relatively small coefficients. The word “automatically” means to indicate that the algorithm can be implemented in Magma. Numerous famous examples are verified and a new theorem is proved by running simply the corresponding Magma procedure requires only the six coefficients of the system a1x 2 + b1y 2...
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ژورنال
عنوان ژورنال: Indagationes Mathematicae
سال: 1997
ISSN: 0019-3577
DOI: 10.1016/s0019-3577(97)81817-4