摘要 :
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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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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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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We consider the relative Thue inequalities vertical bar X-4-t(2)X(2)Y(2)+s(2)Y(4)vertical bar root 550. Moreover we solve the relative Time equations X-4-t(2)X(2)Y(2)+Y-4=mu for vertical bar t vertical bar > root 245, where the p...
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We consider the relative Thue inequalities vertical bar X-4-t(2)X(2)Y(2)+s(2)Y(4)vertical bar root 550. Moreover we solve the relative Time equations X-4-t(2)X(2)Y(2)+Y-4=mu for vertical bar t vertical bar > root 245, 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 solve these Time inequalities respectively equations by using the method of Thue-Siegel.
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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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The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known, see G. Lettl, A. Petho and P. Voutier, Simple families of Thue inequalities, Trans. Amer. Math. Soc. 351 (1...
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The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known, see G. Lettl, A. Petho and P. Voutier, Simple families of Thue inequalities, Trans. Amer. Math. Soc. 351 (1999) 1871-1894, On the arithmetic of simplest sextic fields and related Thue equations, in Number Theory: Diophantine, Computational and Algebraic Aspects, eds. K. Gyory et al. (de Gruyter, Berlin, 1998), pp. 331-348. The family of simplest cubic Thue equations was already studied in the relative case, over imaginary quadratic fields. In the present paper, we give a similar extension of simplest quartic and simplest sextic Thue equations over imaginary quadratic fields. We explicitly give the solutions of these infinite parametric families of Thue equations over arbitrary imaginary quadratic fields.
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摘要 :
Diophantine equations can often be reduced to various types of classical Thue equations. These equations usually have only very small solutions. On the other hand, to compute all solutions (i.e., to prove the nonexistence of large...
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Diophantine equations can often be reduced to various types of classical Thue equations. These equations usually have only very small solutions. On the other hand, to compute all solutions (i.e., to prove the nonexistence of large solutions) is a time-consuming procedure. Therefore, it is useful to have a fast algorithm to calculate the "small" solutions, especially if "small" means less than, e.g., 10100. Such an algorithm was constructed by A. Petho in 1987 based on continued fractions.
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