Geodesic
A Further Generalization of an Irreducibility Theorem of A. Cohn
Canadian journal of mathematics, Tome 34 (1982) no. 6, pp. 1390-1395

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Let d n dn–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 ≦ dk ≦ 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. 
Filaseta, Michael. A Further Generalization of an Irreducibility Theorem of A. Cohn. Canadian journal of mathematics, Tome 34 (1982) no. 6, pp. 1390-1395. doi: 10.4153/CJM-1982-097-3
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[1] 1. Brillhart, J., Filaseta, M. and Odlyzko, A., On an irreducibility theorem of A. Cohn, Can. J. Math. 33 (1981). Google Scholar

[2] 2. Polya, G. and Szego, G., Aufgaben und Lehrsdtze aus der Analysis (Springer-Verlag, Berlin, 1964). Google Scholar

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