Determination of Bounds for the Solutions to those Binary Diophantine Equations that Satisfy the Hypotheses of Runge's Theorem

Hilliker, David Lee; Straus, E. G.

doi:10.2307/1999638

Cited by 17 publications

(34 citation statements)

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“…Some of the arguments in our proof, particularly those involving Puiseux series, follow arguments given in [7] and [30].…”

Section: Variations On a Theme Of Rungementioning

confidence: 80%

Variations on a theme of Runge: effective determination of integral points on certain varieties

Levin¹

2008

J. Théor. Nombres Bordeaux

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We consider some variations on the classical method of Runge for effectively determining integral points on certain curves. We first prove a version of Runge's theorem valid for higher-dimensional varieties, generalizing a uniform version of Runge's theorem due to Bombieri. We then take up the study of how Runge's method may be expanded by taking advantage of certain coverings. We prove both a result for arbitrary curves and a more explicit result for superelliptic curves. As an application of our method, we completely solve certain equations involving squares in products of terms in an arithmetic progression.

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“…Some of the arguments in our proof, particularly those involving Puiseux series, follow arguments given in [7] and [30].…”

Section: Variations On a Theme Of Rungementioning

confidence: 80%

Variations on a theme of Runge: effective determination of integral points on certain varieties

Levin¹

2008

J. Théor. Nombres Bordeaux

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“…We provide bounds for the size of solutions and an algorithm to determine all solutions (x, y) ∈ Z 2 . The method of proof is based on Runge's method [15,21,27,31,34,37]. In 2008, Sankaranarayanan and Saradha [28] established improved upper bounds for the size of the solutions of the Diophantine equations F (x) = y m and F (x) = G(y), for which Runge's method can be applied.…”

Section: Introductionmentioning

confidence: 99%

On a generalization of a problem of Erdos and Graham

Tengely¹,

Varga²

2014

Publ. Math. Debrecen

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“…Runge's method of proof is effective, that is, it yields computable upper bounds for the sizes of the integer solutions to these equations. Using this method upper bounds were obtained by Hilliker and Straus [8] and by Walsh [20]. Grytczuk and Schinzel [6] applied a method of Skolem [17] based on elimination theory to obtain upper bounds for the solutions.…”

Section: Introduction Consider a Polynomialmentioning

confidence: 99%

On the Diophantine equation F(x)=G(y)

Tengely

2003

Acta Arith.

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Determination of Bounds for the Solutions to those Binary Diophantine Equations that Satisfy the Hypotheses of Runge's Theorem

Cited by 17 publications

References 0 publications

Variations on a theme of Runge: effective determination of integral points on certain varieties

Variations on a theme of Runge: effective determination of integral points on certain varieties

On a generalization of a problem of Erdos and Graham

On the Diophantine equation F(x)=G(y)

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