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A Further Generalization of an Irreducibility Theorem of A. Cohn

1982; Cambridge University Press; Volume: 34; Issue: 6 Linguagem: Inglês

10.4153/cjm-1982-097-3

1496-4279

Michael Filaseta,

Polynomial and algebraic computation

Let d n d n –1 … d 0 be the b -ary representation of a positive integer N . Call the polynomial obtained from N base b . In the case the base is 10, f ( x ) will be called the polynomial obtained from N . Pólya and Szegö attribute the following theorem to A. Cohn [ 2 , b. 2, VIII, 128]: THEOREM 1. A polynomial obtained from a prime is irreducible. This theorem was generalized in two different ways by John Brillhart, Andrew Odlyzko, and myself [ 1 ]. One way was by proving the theorem remains true regardless of the base being used. The second way was by permitting the coefficients of f ( x ) to be different from digits. Thus, for example, if , where 0 ≦ d k ≦ 167 for all k , and if f (10) is prime, then f ( x ) is irreducible. In this paper, Theorem 1 will be generalized in another way by considering composite N .