What question did this study set out to answer?

To investigate the conditions under which elementary symmetric functions of the set {1, 1/2, ..., 1/n} without 1/i yield integer values.

May 27, 2026Open Access

On the elementary symmetric functions of 1, 1/2,. . . , 1/n\1/i

Key Points

To investigate the conditions under which elementary symmetric functions of the set {1, 1/2, ..., 1/n} without 1/i yield integer values.
Proofs build upon past works by Erdős and Niven (1946) and Chen and Tang (2012).
Examination of cases for n values greater than or equal to 5 excluding specific integers.
Logical arguments based on properties of symmetric functions and number theory.
Confirmed that none of the elementary symmetric functions of {1, 1/2,..., 1/n}\{1/i} are integers for n ≥ 5, except specific known exceptions (n=2 and n=4).

Abstract

In 1946, P. Erdős and I. Niven proved that there are only finitely many positive integers n for which one or more of the elementary symmetric functions of 1, 1/2,. . . , 1/n are integers. In 2012, Y. Chen and M. Tang proved that if n ⩾ 4, then none of the elementary symmetric functions of 1, 1/2,. . . , 1/n are integers. In this paper, we prove that if n ⩾ 5, then none of the elementary symmetric functions of 1, 1/2,. . . , 1/n\1/i are integers except for n = i = 2 and n = i = 4.

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