1982
DOI: 10.2307/2007295
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An Algorithm for Solving a Certain Class of Diophantine Equations. I

Abstract: Abstract. A class of Diophantine equations is defined and an algorithm for solving each equation in this class is developed. The methods consist of techniques for the computation of an upper bound for the absolute value of each solution. The computability of these bounds is guaranteed. Typically, these bounds are well within the range of computer programming and so they constitute a practical method for computing all solutions to the Diophantine equation in question. As a first application, a bound for a cubic… Show more

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Cited by 3 publications
(5 citation statements)
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“…The main result of current work is the algorithm for solving the equation 3and an estimation of its complexity. Moreover, the proposed algorithm permits to give the significantly better estimates for the solutions of this equation (in comparison to those that the estimates of the type (2) give in our case). The article is organized as follows.…”
Section: Introductionmentioning
confidence: 79%
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“…The main result of current work is the algorithm for solving the equation 3and an estimation of its complexity. Moreover, the proposed algorithm permits to give the significantly better estimates for the solutions of this equation (in comparison to those that the estimates of the type (2) give in our case). The article is organized as follows.…”
Section: Introductionmentioning
confidence: 79%
“…Moreover, the upper bound H + √ 2H 2 + 1 is achieved for these (infinitely many) H, for which the number √ 2H 2 + 1 is integer. Earlier in the paper [2], the less accurate estimate |x| < 10H was given.…”
Section: Remark 2 For the Equation (3)mentioning
confidence: 99%
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“…Hilliker [3,4] has developed techniques for solving certain Diophantine equations. To illustrate those techniques the quartic case of Runge's Theorem has been developed (Hilliker [4]).…”
Section: =07=0mentioning
confidence: 99%
“…To illustrate those techniques the quartic case of Runge's Theorem has been developed (Hilliker [4]). The methods of the papers [3,4] are different from those of the present paper, but both approaches rest upon certain numerical techniques in the classical theory of algebraic functions.…”
Section: =07=0mentioning
confidence: 99%