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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 \((n,k)\).

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