Simultaneous inequalities among values of the Euler phi-function
| dc.creator | Martin, Greg | |
| dc.date | 2006-03-02 | |
| dc.date.accessioned | 2026-07-07T07:06:27Z | |
| dc.date.available | 2026-07-07T07:06:27Z | |
| dc.description | This paper concerns the values of the Euler phi-function evaluated simultaneously on k arithmetic progressions a_1 n + b_1, a_2 n + b_2, ..., a_k n + b_k. Assuming the necessary condition that no two of the polynomials a_i x + b_i are constant multiples of each other, we show that there are infinitely many integers n for which phi(a_1 n + b_1) > phi(a_2 n + b_2) > ... > phi(a_k n + b_k). In particular, there exist infinitely many strings of k consecutive integers whose phi-values are arranged from largest to smallest in any prescribed manner. Also, under the necessary condition ad \ne bc, any inequality of the form phi(an+b) < phi(cn+d) infinitely often has k consecutive solutions. In fact, we prove that the sets of solutions to these inequalities have positive lower density. | |
| dc.description | 8 pages | |
| dc.identifier | https://arxiv.org/abs/math/0603053 | |
| dc.identifier | http://arxiv.org/abs/math/0603053 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/110033 | |
| dc.subject | Number Theory | |
| dc.subject | 11A25 | |
| dc.title | Simultaneous inequalities among values of the Euler phi-function | |
| dc.type | text |