| Автор | Boyd, David W. |
| Дата выпуска | 1994 |
| dc.description | Let H <sub> n </sub> = 1 + ½ + … + be the n-th partial sum of the harmonic series. A classical result of Wolstenholme states that, if p > 3 is prime, the numerator of H <sub> p </sub>–l is divisible by p <sup>2</sup>. Here we consider, for a given prime p, the set J <sub> p </sub> of n for which p divides the numerator of H <sub> n </sub>. This set J <sub> p </sub> had been previously determined for p = 2,3,5,7. One of our results is that J <sub>11</sub> contains exactly 638 integers, the largestof which is a number of 31 decimal digits. We determine J <sub> p </sub> for all p < 550 with three exceptions: 83, 127 and 397.The computation is based on a new p-adically convergent formula for the quantity H <sub> pn </sub> – H <sub> n </sub>/p. We describe a probabilistic model for the sets J <sub> p </sub>, based on branching processes. The model predicts that |J <sub> p </sub>| = O(p <sup>2</sup>(log log p)<sup>2+∊</sup>), and that there are infinitely many p with |J <sub> p </sub>| ≥ p <sup>2</sup>(log log p)<sup>2</sup>. This strengthens an earlier conjecture of Eswarathasan and Levine that |J <sub>p</sub>| is finite for all p. Another prediction of the model is that there will be infin itely many pairs (n,p) for which p <sup>3</sup> divides the numerator of H <sub> n </sub>, but only finitely many for which p <sup>4</sup> divides H <sub> n </sub>.It has been conjectured that there are infinitely many p for which |J<sub>p</sub> | = 3. We give a probabilistic argument that suggests that such primes have a density 1/e in the set of all primes, and experimentally confirm this by a determination of all such p ≤ 10<sup>5</sup>. |
| Формат | application.pdf |
| Издатель | Taylor & Francis Group |
| Копирайт | Copyright Taylor and Francis Group, LLC |
| Название | A p-adic Study of the Partial Sums of the Harmonic Series |
| Тип | research-article |
| DOI | 10.1080/10586458.1994.10504298 |
| Electronic ISSN | 1944-950X |
| Print ISSN | 1058-6458 |
| Журнал | Experimental Mathematics |
| Том | 3 |
| Первая страница | 287 |
| Последняя страница | 302 |
| Аффилиация | Boyd, David W.; Department of Mathematics, University of British Columbia |
| Выпуск | 4 |
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