Prime values of reducible polynomials, II
| dc.creator | Chen, Yong-Gao | |
| dc.creator | Kun, Gabor | |
| dc.creator | Pete, Gabor | |
| dc.creator | Ruzsa, Imre Z. | |
| dc.creator | Timar, Adam | |
| dc.date | 2005-10-17 | |
| dc.date.accessioned | 2026-07-07T06:47:34Z | |
| dc.date.available | 2026-07-07T06:47:34Z | |
| dc.description | The Schinzel hypothesis claims (but it seems hopeless to prove) that any irreducible Q[x] polynomial without a constant factor assumes infinitely many prime values at integer places. On the other hand, it is easy to see that a reducible Q[x] polynomial can have only finitely many such places. In this paper we prove that a reducible Z[x] polynomial of degree n (where n is not 4 or 5) can have at most n+2 such places, and there exist examples with n+1 places. If the Schinzel hypothesis and the k-prime-tuple conjecture are true, then there are also polynomials with n+2 such places. (For n=4 or 5 the maximum possible value is 8.) If a Z[x] polynomial is the product of two non-constant integer-valued Q[x] polynomials, then there are at most 1.87234...n+o(n) such places. Even in this case, the number of integer places with positive prime values is at most n. More generally, if f=gh \in R[x] is a product of two non-constant real polynomials, then the number of real places x such that |g(x)|=1 or |h(x)=1|, while f(x)>1, is at most n. For a natural complex version of this last statement we give a counterexample. Furthermore, we briefly consider the generalized problem of assuming values in a fixed finite set of integers. | |
| dc.description | 12 pages | |
| dc.identifier | https://arxiv.org/abs/math/0510357 | |
| dc.identifier | http://arxiv.org/abs/math/0510357 | |
| dc.identifier | Acta Arithmetica, 104 (2002), no. 2, 117--127 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/103697 | |
| dc.subject | Number Theory | |
| dc.subject | 11N32 | |
| dc.title | Prime values of reducible polynomials, II | |
| dc.type | text |