Erdős and Niven (1942)- Integrality of Harmonic Sums

THEOREM: There is only a finite number of integers n . for which one
or more of the elementary symmetric functions of \displaystyle 1,1 / 2,1 / 3, \cdots, 1 / n is an integer.

Proof: For k small enough (k < 3 \log n ), the k-th elementary symmetric function of the \displaystyle 1,1 / 2,1 / 3, \cdots, 1 / n is less than \displaystyle \frac{(1+1 / 2+\cdots+1 / n)^{k}}{k !}<\frac{(1+\log n)^{k}}{k !}<1.
For larger k , use existence of primes in short intervals [x, x+x^{5/8}] to find a prime in the interval [1+n /(k+1), n / k]. then \displaystyle p, 2p, \cdots kp <n and contribute a term to the k-symmetric function with a denominator p. 2p.3p.\cdots kp=k!p^k – this is the only denominator which is divisible by p^k and hence once you take common denominators- we see that denominator is divisible by p and the numerator is not. \square

What about more general fractions?- denominators in arithmetic progressions, some coprime numerators, more general polynomial sequences for denominators etc?

https://projecteuclid.org/download/pdf_1/euclid.bams/1183507840

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