Binomial Thue equations and polynomial powers

Bennett  MA; Gyory  K; Mignotte  MPinter  A

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Binomial Thue equations and polynomial powers

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We explicitly solve a collection of binomial Thue equations with unknown degree and unknown S-unit coefficients, for a number of sets S of small cardinality. Equivalently, we characterize integers x such that the polynomial x(2) + x assumes perfect power values, modulo S-units. These results are proved through a combination of techniques, including Frey curves and associated modular forms, lower bounds for linear forms in logarithms, the hypergeometric method of Thue and Siegel, local methods, and computational approaches to Thue equations of low degree. Along the way, we derive some new results on Fermat-type ternary equations, combining classical cyclotomy with Frey curve techniques.

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来源
《Compositio mathematica》 |2006年第5期|1103-1121|共19页
作者
Bennett MA; Gyory K; Mignotte MPinter A;
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作者单位

Univ British Columbia, Dept Math, Vancouver, BC, Canada;

Univ Debrecen, Hungarian Acad Sci, Number Theory Res Grp, Math Inst, Debrecen, Hungary;

Univ Strasbourg, Dept Math, Strasbourg, Franceri.kyoto-u.ac.jp;

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正文语种英语
中图分类组合数学（组合学）;
关键词
binomial Thue equations; superelliptic equations; explicit resolution; TERNARY DIOPHANTINE EQUATIONS; FERMATS LAST THEOREM; PERFECT POWERS; CONSECUTIVE INTEGERS; ALGEBRAIC-NUMBERS; FORMS; REPRESENTATIONS; LOGARITHMS; CURVES; VALUES;

Binomial Thue equations and polynomial powers

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