摘要 :
In Part I (cf. [13]) of this paper, the title equation was solved in x, y, n ? Z with |xy| > 1, n ≥ 3 for a collection of positive integers A, B, C under certain bounds. In the present paper we extend these results to much larger...
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In Part I (cf. [13]) of this paper, the title equation was solved in x, y, n ? Z with |xy| > 1, n ≥ 3 for a collection of positive integers A, B, C under certain bounds. In the present paper we extend these results to much larger ranges of A, B, C. We give among other things all the solutions for A = C = 1, B < 235 (cf. Theorem 1), and for C= 1, A, B ≤ 50, with six explicitly given exceptions (A, B, n) (cf. Theorem 3). The equations under consideration are solved by combining powerful techniques, including Prey curves and associated modular forms, lower bounds for linear forms in logarithms, the hypergeometric method of Thue and Siegel, local methods, classical cyclotomy and computational approaches to Thue equations of low degree. Along the way, we derive a new result on the solvability of binomial Thue equations (cf. Theorem 6) which is crucial in the proof of our Theorems 1 and 2. Some important applications of our theorems will be given in a forthcoming paper.
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摘要 :
We give a rather sharp upper bound for the degree of a binomial Thue--Mahler equation in terms of the coefficients and the primes involved. Further, we establish explicit lower bounds for the greatest prime factor of a binomial bi...
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We give a rather sharp upper bound for the degree of a binomial Thue--Mahler equation in terms of the coefficients and the primes involved. Further, we establish explicit lower bounds for the greatest prime factor of a binomial binary form at integral points. Our estimates considerably generalize and improve the earlier results obtained in this direction.
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摘要 :
The classical equations (1) and (3) have a very extensive literature. The main purpose of recent investigations is to solve these equations with as large bound for the greatest prime factor P(b) of b as possible. General elementar...
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The classical equations (1) and (3) have a very extensive literature. The main purpose of recent investigations is to solve these equations with as large bound for the greatest prime factor P(b) of b as possible. General elementary methods have been developed for studying (1) and (3) which, however, cannot be applied if k is small. As a generalization of previous results obtained for small values of k, we completely solve Eq. (1) for k ≤ 5, under the assumption that P(b) ≤ p k , the k-th prime (cf. Theorem 1). A similar result is established for Eq. (3) (cf. Theorem 2). In our proofs, several deep results and powerful techniques are combined from modern Diophantine analysis.
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摘要 :
The classical equations (1) and (3) have a very extensive literature. The main purpose of recent investigations is to solve these equations with as large bound for the greatest prime factor P(b) of b as possible. General elementar...
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The classical equations (1) and (3) have a very extensive literature. The main purpose of recent investigations is to solve these equations with as large bound for the greatest prime factor P(b) of b as possible. General elementary methods have been developed for studying (1) and (3) which, however, cannot be applied if k is small. As a generalization of previous results obtained for small values of k, we completely solve Eq. (1) for k 收起
摘要 :
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) +...
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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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