Glasnik matematički, Vol. 45 No. 1, 2010.
Original scientific paper
https://doi.org/10.3336/gm.45.1.03
On the reducibility of certain quadrinomials
Jonas Jankauskas
orcid.org/0000-0001-9770-7632
; Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania
Abstract
In 2007 West Coast Number Theory conference Walsh asked to determine all irreducible polynomials of the form P(x) = xi + xj + xk + 4 with integer exponents i > j > k > 0, such that for some positive integer l the polynomial P(xl) is reducible in Z[x]. In this paper we prove that such polynomials are quadrinomials x4m + x3m + x2m + 4, where m is an odd positive integer. In addition, Walsh asked for the examples of reducible quadrinomials xi + xj + xk + n, n > 4 with no linear or quadratic factors. We compute the examples of reducible polynomials of the form above with non-trivial factors and negative coefficient n.
Keywords
Hrčak ID:
52365
URI
Publication date:
17.5.2010.
Visits: 1.407 *