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On Rational Approximations by Pythagorean Number !
| Content Provider | Semantic Scholar |
|---|---|
| Author | Eisner, Carsten |
| Copyright Year | 2000 |
| Abstract | hold ([2], Theorem 225). Moreover, there is a (1,1) correspondence between different values of o, b and different values of x, y, z. The object of this paper is to investigate diophantine inequalities \£y — x\ for integers y and x from triplets of Pythagorean numbers. Since x + y is required to be a perfect square in what follows we write x + y € • we have a essential restriction on the rationals x/y approximating a real irrational £. So one may not expect to get a result as strong as Heilbronn's theorem. Indeed, there are irrationals £ such that \iy — x\ ^$> 1 holds for all integers x, y satisfying x + y E D . But almost all real irrationals £ (in the sense of the Lebesgue-measure) can be approximated in such a way that |fy — x\ tends to zero for a infinite sequence or pairs x, ycorresponding to Pythagorean numbers. In order to prove our results we shall make use of the properties of continued fraction expansions. By our first theorem we describe those real irrationals having good approximations by Pythagorean numbers. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.mathstat.dal.ca/FQ/Scanned/41-2/elsner.pdf |
| Alternate Webpage(s) | http://www.fq.math.ca/Scanned/41-2/elsner.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
