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

Reducibility; quadrinomials

Hrčak ID:

52365

URI

https://hrcak.srce.hr/52365

Publication date:

17.5.2010.

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