Wyniki wyszukiwania

1

On the irreducibility of neighbouring polynomials

100%

Győry K.

Acta Arithmetica

|

1994

|

tom 67

|

nr 3

283-294

2

On the diophantine equation ${n \choose k} = x^l$

100%

Győry K.

Acta Arithmetica

|

1997

|

tom 80

|

nr 3

289-295

EN

P. 294, line 14: For “Satz 8” read “Satz 7”, and for “equation (10)” read “equation (13)”.

3

Some applications of decomposable form equations to resultant equations

100%

Győry K.

Colloquium Mathematicum

|

1993

|

tom 65

|

nr 2

267-275

EN

1. Introduction. The purpose of this paper is to establish some general finiteness results (cf. Theorems 1 and 2) for resultant equations over an arbitrary finitely generated integral domain R over ℤ. Our Theorems 1 and 2 improve and generalize some results of Wirsing [25], Fujiwara [6], Schmidt [21] and Schlickewei [17] concerning resultant equations over ℤ. Theorems 1 and 2 are consequences of a finiteness result (cf. Theorem 3) on decomposable form equations over R. Some applications of Theorems 1 and 2 are also presented to polynomials in R[X] assuming unit values at many given points in R (cf. Corollary 1) and to arithmetic progressions of given order, consisting of units of R (cf. Corollary 2). Further applications to irreducible polynomials will be given in a separate paper. Our Theorem 3 seems to be interesting in itself as well. It is deduced from some general results of Evertse and the author [3] on decomposable form equations. Since the proofs in [3] depend among other things on the Thue-Siegel-Roth-Schmidt method and its p-adic generalization

4

Upper bounds for the degrees of decomposable forms of given discriminant

100%

Győry K.

Acta Arithmetica

|

1994

|

tom 66

|

nr 3

261-268

EN

1. Introduction. In our paper [5] a sharp upper bound was given for the degree of an arbitrary squarefree binary form F ∈ ℤ[X,Y] in terms of the absolute value of the discriminant of F. Further, all the binary forms were listed for which this bound cannot be improved. This upper estimate has been extended by Evertse and the author [3] to decomposable forms in n ≥ 2 variables. The bound obtained in [3] depends also on n and is best possible only for n = 2. The purpose of the present paper is to establish an improvement of the bound of [3] which is already best possible for every n ≥ 2. Moreover, all the squarefree decomposable forms in n variables over ℤ will be determined for which our bound cannot be further sharpened. In the proof we shall use some results and arguments of [5] and [3] and two theorems of Heller [6] on linear systems with integral valued solutions.

5

On the irreducibility of a class of polynomials, IV

100%

Győry K.

Acta Arithmetica

|

1992

|

tom 62

|

nr 4

399-405

6

On the diophantine equation $n(n+1)...(n+k-1) = bx^l$

100%

Győry K.

Acta Arithmetica

|

1998

|

tom 83

|

nr 1

87-92

7

On the abc conjecture in algebraic number fields

100%

Győry K.

Acta Arithmetica

|

2008

|

tom 133

|

nr 3

281-295

8

Effective finiteness theorems for decomposable forms of given discriminant

64%

Evertse J., Győry K.

Acta Arithmetica

|

1991-1992

|

tom 60

|

nr 3

233-277

9

On prime factors of integers of the form (ab+1)(bc+1)(ca+1)

64%

Győry K., Sárközy A.

Acta Arithmetica

|

1997

|

tom 79

|

nr 2

163-171

EN

1. Introduction. For any integer n > 1 let P(n) denote the greatest prime factor of n. Győry, Sárközy and Stewart [5] conjectured that if a, b and c are pairwise distinct positive integers then (1) P((ab+1)(bc+1)(ca+1)) tends to infinity as max(a,b,c) → ∞. In this paper we confirm this conjecture in the special case when at least one of the numbers a, b, c, a/b, b/c, c/a has bounded prime factors. We prove our result in a quantitative form by showing that if 𝓐 is a finite set of triples (a,b,c) of positive integers a, b, c with the property mentioned above then for some (a,b,c) ∈ 𝓐, (1) is greater than a constant times log|𝓐|loglog|𝓐|, where |𝓐| denotes the cardinality of 𝓐 (cf. Corollary to Theorem 1). Further, we show that this bound cannot be replaced by $|𝓐|^ε$ (cf. Theorem 2). Recently, Stewart and Tijdeman [9] proved the conjecture in another special case. Namely, they showed that if a ≥ b > c then (1) exceeds a constant times log((loga)/log(c+1)). In the present paper we give an estimate from the opposite side in terms of a (cf. Theorem 3).

10

Counting solutions of decomposable form equations

64%

Everest G., Győry K.

Acta Arithmetica

|

1997

|

tom 79

|

nr 2

173-191

11

The number of families of solutions of decomposable form equations

64%

Evertse J., Győry K.

Acta Arithmetica

|

1997

|

tom 80

|

nr 4

367-394

12

Integer solutions of a sequence of decomposable form inequalities

64%

Győry K., Ru M.

Acta Arithmetica

|

1998

|

tom 86

|

nr 3

227-237

13

Equal values of binary forms at integral points

52%

Evertse J.-H., Győry K., Shorey T. N., Tijdeman R.

Acta Arithmetica

|

1987

|

tom 48

|

nr 4

379-396

14

On the number of prime factors of integers of the form ab + 1

52%

Győry K., Sárközy A., Stewart C.

Acta Arithmetica

|

1996

|

tom 74

|

nr 4

365-385

15

On prime factors of sums of integers III

39%

Győry K., Stewart C. L., Tijdeman R.

Acta Arithmetica

|

1987-1988

|

tom 49

|

nr 3

307-312

16

Finiteness criteria for decomposable form equations

32%

Evertse J.-H., Győry K.

Acta Arithmetica

|

1988

|

tom 50

|

nr 4

357-379

17

On some arithmetical properties of Lucas and Lehmer numbers

32%

Győry K.

Acta Arithmetica

|

1981-1982

|

tom 40

|

nr 4

369-373

Ograniczanie wyników

On the irreducibility of neighbouring polynomials

On the diophantine equation ${n \choose k} = x^l$

Some applications of decomposable form equations to resultant equations

Upper bounds for the degrees of decomposable forms of given discriminant

On the irreducibility of a class of polynomials, IV

On the diophantine equation $n(n+1)...(n+k-1) = bx^l$

On the abc conjecture in algebraic number fields

Effective finiteness theorems for decomposable forms of given discriminant

On prime factors of integers of the form (ab+1)(bc+1)(ca+1)

Counting solutions of decomposable form equations

The number of families of solutions of decomposable form equations

Integer solutions of a sequence of decomposable form inequalities

Equal values of binary forms at integral points

On the number of prime factors of integers of the form ab + 1

On prime factors of sums of integers III

Finiteness criteria for decomposable form equations

On some arithmetical properties of Lucas and Lehmer numbers