A generalization of a second irreducibility theorem of I. Schur

A generalization of a third irreducibility theorem of I. Schur

“…This work is continuation of earlier work by the authors in [5] and [1] in which the role of x 2j /u 2j above is replaced by x j /j! and x j /(j + 1)!, respectively.…”

“…The Newton polygon N G (7) consists of two edges, one from (0, 0) to (2, 0) and one from (2, 0) to (10,1). By (C), (D) and (E), we obtain 7 | a 5 .…”

“…The conditions on a n were different for these results, but in a manner similar to that just described, the results there were best possible. The general techniques used for establishing Theorem 1 are similar to those used in [1], [5] and [6]. The authors were not, however, able to take advantage of work by E. F. Ecklund, Jr., R. B. Eggleton, P. Erdős, and J. L. Selfridge [3] that played a crucial role in the prior two papers [5] and [1] on the subject.…”

“…The general techniques used for establishing Theorem 1 are similar to those used in [1], [5] and [6]. The authors were not, however, able to take advantage of work by E. F. Ecklund, Jr., R. B. Eggleton, P. Erdős, and J. L. Selfridge [3] that played a crucial role in the prior two papers [5] and [1] on the subject. We note that there were four irreducibility theorems of I. Schur contained in [9] and [10]; the authors plan to address the fourth of these in a subsequent paper.…”

“…In the prior work of [1] and [5], we were able to take advantage of a result by Ecklund, Eggleton, Erdős, and Selfridge [3] in which a similar product is considered over p dividing instead a product of k consecutive integers. We are no longer able to appeal to this result and instead establish Lemma 7 based on other estimates in the distribution of primes.…”