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In this paper we consider the family of systems (2c + 1)U~2 — 2cV~2 = μ and (c — 2)U~2 — cZ~2 = — 2μ of relative Pellian equations, where the parameter c and the root of unity μ are integers in the same imaginary quadratic ...
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In this paper we consider the family of systems (2c + 1)U~2 — 2cV~2 = μ and (c — 2)U~2 — cZ~2 = — 2μ of relative Pellian equations, where the parameter c and the root of unity μ are integers in the same imaginary quadratic number field K = Q((-d)~(1/2)). We show that for |c| ≥ 3 only certain values of μ yield solutions of this system, and solve the system completely for |c| ≥ 1544686. Furthermore we will consider the related relative Thue equation X~4 - 4cX~3Y + (6c + 2)X~2Y~2 + 4cXY~3 + Y~4 = μ and solve it by the method of Tzanakis under the same assumptions.
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Let F(x, y) be an irreducible binary form of degree ≥ 3 with integer coefficients and with real roots. Let M be an imaginary quadratic field with ring of integers ZM. Let K > 0. We describe an efficient method how to reduce the r...
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Let F(x, y) be an irreducible binary form of degree ≥ 3 with integer coefficients and with real roots. Let M be an imaginary quadratic field with ring of integers ZM. Let K > 0. We describe an efficient method how to reduce the resolution of the relative Thue inequalities |F(x,y)| ≤K (x,y ∈ Zm) to the resolution of absolute Thue inequalities of type |F(x,y)| ≤k (x,y∈Z). We illustrate our method with an explicit example.
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Let F_n be a binary form with integral coefficients of degree n≥2, let d denote the greatest common divisor of all non-zero coefficients of F_n, and let h≥2 be an integer. We prove that if d=1 then the Thue equation (T) F_n(x,y)...
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Let F_n be a binary form with integral coefficients of degree n≥2, let d denote the greatest common divisor of all non-zero coefficients of F_n, and let h≥2 be an integer. We prove that if d=1 then the Thue equation (T) F_n(x,y)=h has relatively few solutions: if A is a subset of the set T(F_n,h) of all solutions to (T), with r:=card(A)≥n+1, then. (#)h divides the number Δ(A):=∏1≤k收起
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For an integral parameter t is an element of Z we investigate the family of Thue equations F(x, y) = x(5) + (t - 1)(2)x(4)y - (2t(3) + 4t + 4)x(3)y(2) + (t(4) + t(3) + 2t(2) + 4t - 3)x(2)y(3) + (t(3) + t(2) + 5t + 3)xy(4) + y(5) =...
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For an integral parameter t is an element of Z we investigate the family of Thue equations F(x, y) = x(5) + (t - 1)(2)x(4)y - (2t(3) + 4t + 4)x(3)y(2) + (t(4) + t(3) + 2t(2) + 4t - 3)x(2)y(3) + (t(3) + t(2) + 5t + 3)xy(4) + y(5) = +/-1, originating from Emma Lehmer's family of quintic fields, and show that for \t\ greater than or equal to 3.28.10(15) the only solutions are the trivial ones with x = 0 or y = 0. Our arguments contain some new ideas in comparison with the standard methods for Thue families, which gives this family a special interest. [References: 13]
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We completely solve the family of relative Thue equations x~3 - (t - 1)x~2y -(t + 2)xy~2 - y~3 = μ where the parameter t, the root of unity μ and the solutions x and y are integers in the same imaginary quadratic number field. T...
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We completely solve the family of relative Thue equations x~3 - (t - 1)x~2y -(t + 2)xy~2 - y~3 = μ where the parameter t, the root of unity μ and the solutions x and y are integers in the same imaginary quadratic number field. This is achieved using the hypergeometric method for |t| ≥ 53 and Baker's method combined with a computer search using continued fractions for the remaining values of t.
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摘要 :We consider the relative Thue equations \[X^3 - t X^2 Y - (t+1) X Y^2 -Y^3=\mu,\] where the parameter ]]>]]>]]>]]>]]>t$, the root of unity $\mu$ and the solutions $X$ and $Y$ are integers in the same imaginary quadratic number fie...
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We consider the relative Thue equations \[X^3 - t X^2 Y - (t+1) X Y^2 -Y^3=\mu,\] where the parameter ]]>]]>]]>]]>]]>t$, the root of unity $\mu$ and the solutions $X$ and $Y$ are integers in the same imaginary quadratic number field. We use Baker's method to find all solutions for $|t|> 2.88 \cdot 10^{33}$.
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A theorem of Rihane, Hernane and Togbe on a parametric family of simultaneous Pell equations is proved using classical results on quartic Diophantine equations due to Cohn and to Ljunggren.
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We consider the problem of counting solutions to a trinomial Thue equation - that is, an equation (*) \F(x, y)\ = 1, where F is an irreducible form in Z[x, y] with degree at least three and with three non-zero coefficients. In a 1...
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We consider the problem of counting solutions to a trinomial Thue equation - that is, an equation (*) \F(x, y)\ = 1, where F is an irreducible form in Z[x, y] with degree at least three and with three non-zero coefficients. In a 1987 paper J. Mueller and W. Schmidt gave effective bounds for this problem. Their work was based on a series of papers by Bombieri, Bombieri-Mueller and Bombieri-Schmidt, all concerned with the "Thue-Siegel principle" and its relation to (*). In this paper we give specific numerical bounds for the number of solutions to (*) by a somewhat different approach, the difference lying in the initial step-solving a certain diophantine approximation problem. We regard this as a real variable extremal problem, which we then solve by elementary calculus. [References: 23]
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We establish upper bounds for the number of primitive integer solutions to inequalities of the shape , where , , , and are algebraic constants with , and and h are integers. As an important application, we pay special attention to...
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We establish upper bounds for the number of primitive integer solutions to inequalities of the shape , where , , , and are algebraic constants with , and and h are integers. As an important application, we pay special attention to binomial Thue's inequalities . The proofs are based on the hypergeometric method of Thue and Siegel and its refinement by Evertse.
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