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On an Irreducibility Theorem of A. Cohn

Published online by Cambridge University Press:  20 November 2018

John Brillhart ,
Michael Filaseta  and
Andrew Odlyzko
John Brillhart
Affiliation:
The University of Arizona, Tucson, Arizona
Michael Filaseta
Affiliation:
Bell Telephone Laboratories, Murray Hill, New Jersey
Andrew Odlyzko
Affiliation:
Bell Telephone Laboratories, Murray Hill, New Jersey
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In [1, b.2, VIII, 128] Pólya and Szegö give the following interesting result of A. Cohn:

THEOREM 1. If a prime p is expressed in the decimal system as

then the polynomial irreducible inZ[x].

The proof of this result rests on the following theorem of Pólya and Szegö [1, b.2, VIII, 127] which essentially states that a polynomial f(x) is irreducible if it takes on a prime value at an integer which is sufficiently far from the zeros of f(x).

THEOREM 2. Let f(x) ∊ Z[x] be a polynomial with the zeros α1, α2, …, αn. If there is an integer b for which f(b) is a prime, f(b – 1) ≠ 0, and for 1 ≦ i ≦ n, then f(x) is irreducible inZ[x].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 5 , 01 October 1981 , pp. 1055 - 1059
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Pólya, G. and Szego, G., Aufgaben und Lehrsatze aus der Analysis, (Springer-Verlag, Berlin, 1964).Google Scholar