Power values of polynomials and binomial Thue--Mahler equations

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 Frey 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.

Section: Auxiliary Resultsmentioning

confidence: 99%

Section: Appendixmentioning

confidence: 99%

See 1 more Smart Citation

On the resolution of equations $Ax^n-By^n=C$ in integers $x,y$ and $n\geq 3$. II.

Bazsó¹,

Bérczes²,

Győry³

2010

“…We note that a wide range of diophantine problems leads to such equations (see e.g. [1], [2], [3], [12], [13], [15], [17], [19], [18]).…”

Section: Proof Of the Theoremmentioning

confidence: 99%

Further computational experiences on norm form equations with solutions forming arithmetic progressions

Bazsó¹

2007

“…Moreover, in the case when n is also unknown, Tijdeman [26] showed that max{|x|, |y|, n} is still effectively bounded for every integer solution (x, y, n) with |xy| > 1 and n ≥ 3. This effective finiteness result has been extended in [11] to the more general situation when, for a finite set of primes S, the numbers A, B, C are unknown S-units rather than fixed, that is all prime factors of A, B and C lie in S.…”

Section: Introductionmentioning

confidence: 99%

On the resolution of equations $Ax^n-By^n=C$ in integers $x,y$ and $n\geq 3$. I.

Győry¹,

Pintér²

2007

In our paper we initiate a systematic treatment for solving the title equation for bounded positive integer coefficients A, B and C. To illustrate our approach we explicitly solve the equation in integers x, y and n with |xy| > 1, n ≥ 3 for a collection of coefficients A, B, C. We first derive, for concrete values of A, B, C ≤ 100, a relatively small upper bound for n, provided that the equation under consideration has no solution with |xy| ≤ 1 (cf. Theorem 1). Then we give among others all the solutions (x, y, n) for C = 1, A, B ≤ 20 (cf. Theorem 3), and for A = C = 1, B ≤ 70 (cf. Theorem 4). Our method, which may, with some effort, be extended to larger values of A, B and C, combines a wide variety of techniques, classical and modern, in Diophantine analysis.