In [7, b.2, VIII, 128] Pólya and Szegö state the following theorem of A. Cohn:
THEOREM 1. Let dndn−x … d0 be the decimal representation of a prime. Then
is irreducible.
Thus, for example, since 1289 is prime, x
3 + 2x
2 + 8x + 9 is irreducible. Brillhart, Odlyzko, and the author generalized Cohn's Theorem in three different directions. As examples of these types of generalizations, we note the following results, the first two of which are special cases of a result in [1] and the third of a result in [3].
THEOREM 2. Let dndn−x … d0 be the base b representation of a prime where b is an integer ≧2. Then
is irreducible.
THEOREM 3. Let
be such that f(10) is prime and 0 ≦ dj
≦ 167 for j = 0, 1, …, n. Then f(x) is irreducible.