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arXiv · 1301.0758

Integral points on rational curves of the form y=(x^2+bx+c)/(x+a); a,b,c integers

Konstantine Zelator
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Abstract

The subject matter of this work is the set of integral points(i.e. points with both coordinates integers) on the graphs of rational functions of the form f(x)=(x^2+bx+c)/(x+a), with a,b,c,being integers.Following the introduction, we establish Proposition1 in Section2. This proposition plays a key role in the proof of Theorem1 in Section5. Proposition1 is proved with the aid of Euclid's lemma and another well known result in number theory; see reference [1].In Sections3 and4, we focus on the special case b^2-4c=0; which implies b=2d and c=d^2, for some integer d. If d and a are distinct; then there are finitely many integral points, parametrically described in Results1 and2. Theorem1 in Sec.5 deals with the general case.Accordingly, if a^2-ab+c is not zero; there are exactly 4N distinct integral points parametrically described.Except in the cases where a^2-ab+c is a perfect square, or minus a perfect square; in which cases thare are exactly 4N-2 distinct integral points. Here N stands for the number of positive divisors, not exceeding the square root of the absolute value of a^2-ab+c. (divisors of that absolute value).The paper concludes with Th.2, which is a direct application of Th.1 in the cases where the above absolute value is 1, p, or p^2; p a prime.

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Konstantine Zelator. 2013-01-02. Integral points on rational curves of the form y=(x^2+bx+c)/(x+a); a,b,c integers. https://arxiv.org/abs/1301.0758

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