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A generalization of a second irreducibility theorem of I.  Schur. (English) Zbl 1019.11005
Let $$f(X)=\sum_{j=0}^n a_jX^j/(j+1)!$$, where $$a_j$$ are integers and $$a_0=\pm 1$$. Let $$k'$$ be the odd part of $$n$$, and let $$k''$$ be the maximal divisor of $$n(n+1)$$ which is prime to $$6$$. The authors show that if $$0<|a_n|<\min\{k',k''\}$$, then $$f$$ is irreducible, and this result is best possible for every $$n\geq 3$$. This generalizes a theorem of I. Schur [Sitzungsber. Akad. Berlin, Phys.-Math. Kl. 1929, 125-136 (1929; JFM 55.0069.03)]. In one of the lemmas it is shown that if $$n\geq 6$$ and $$k\in[3,n/2]$$, then the maximal divisor of $$\binom{n+1}{k}$$, free of prime factors $$\leq k+1$$, exceeds $$n+1$$, except for $$11$$ pairs $$(n,k)$$.

##### MSC:
 11C08 Polynomials in number theory 11N99 Multiplicative number theory 12E05 Polynomials in general fields (irreducibility, etc.)
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