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On an Irreducibility Theorem of A. Cohn

  • John Brillhart (a1), Michael Filaseta (a2) and Andrew Odlyzko (a2)
Extract

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 in Z[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 α 1, α 2, …, αn . If there is an integer b for which f(b) is a prime, f(b – 1) ≠ 0, and for 1 ≦ in, then f(x) is irreducible in Z[x].

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References
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1. Pólya, G. and Szego, G., Aufgaben und Lehrsatze aus der Analysis, (Springer-Verlag, Berlin, 1964).
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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