A good day to everyone!
I have a (somewhat) intriguing question regarding Paul Erdős's papers on Egyptian fractions (e.g., his 2nd paper during his mathematical career was about this topic).
My question is: What was his (primary) motivation for pursuing this research topic?
In case no historian nor book on the history of mathematics can provide us a definite answer, will it be safe to assume that Erdős was trying to build up a whole theory (and comprehensive framework) to solve the Odd Perfect Number (OPN) conjecture, given the trend in his succeeding research topics in elementary number theory?
I quote Melvin Nathanson who, in his book titled "Elementary Methods in Number Theory", described Erdős as being a "master of elementary methods". In particular, Erdős was able to come up with a proof of the Prime Number Theorem using a purely elementary approach - although, this did not equate to a comparatively simple and/or easy proof.
Prior to the full proof of the PNT, the best estimates for $\pi(n)$ came from Chebyshev-type bounds. Chebyshev (1852) published the bounds
$$0.921292 \frac{x}{\log x} \ll \pi(x) \ll 1.10555\frac{x}{\log x},$$
as $x \to \infty$, and the constants given here were later improved by Sylvester. Then 1892 came, and Chebyshev's techniques were for the most part forgotten.
Around 1937, Erdős showed that Chebyshev-type methods had the potential to prove the PNT, in that the constants above could be taken arbitrarily close to $1$ in the line above. This came with a caveat, though, in that Erdős had assumed the PNT to reach this result (so it was, for the most part, an academic exercise).
In particular, Erdős used an equivalent form of the PNT - that the function
$$ M(n):=\sum_{k \leq n} \mu(n)$$ is $o(n)$. To see where this might come up, recall that Chebyshev obtained the above bounds by consideration of the function
$$ \chi(t):= \psi(t)-\frac{1}{2} \psi(t)-\frac{1}{3}\psi(t)-\frac{1}{5}\psi(t)+\frac{1}{30}\psi(t).$$
Regarding the coefficients, we note that
As for the term $1/30$, this is only used to satisfy (1). Erdős tweaked Chebyshev's idea by considering variants of the $\chi$ function that have general rational coefficients, and it is around here that the growth of the $\mu$ function must be considered.
Disclaimer - Part of the following (the non-historical bits) is conjecture. All the same, it represents my experience with Erdős's work on the elementary PNT:
The set $S=\{1,-2,-3,-5\}$ is very special in that the harmonic sum of $S$ is the reciprocal of an integer. In general, the set $$S_k=\{1,\mu(2),\mu(3),\ldots,\mu(k)\}$$ will not have this property, and we can ask just how many non-zero integers must be added to $S_k$ such that the resulting set has $0$ harmonic sum. (To avoid trivializing this question, we should assume that the new integers $\{n_i\}$ satisfy $\vert n_i \vert >k$.)
If this problem had a nice solution, Erdős would have likely found a novel proof of the PNT. Unfortunately, this problem is difficult, and it is difficult precisely because Egyptian fractions are still poorly understood.
This obstruction could have easily suggested to Erdős the importance of Egyptian fractions in number theory.
Here are some places you may wish to pursue for further reading:
P. L. Chebyshev, Mémoire sur les Nombres Premiers, J. Math. Pures Appl., 17, (1852). (This is Chebyshev's original paper on the topic.)
H. Diamond and P. Erdős, On Sharp Elementary Prime Number Estimates, L’Enseignement Mathématique, 26, (1980). (This includes Erdős's work on the Chebyshev method, in relation to previously established proofs of the PNT.)
A recent blog post of mine, in which Chebyshev's method is generalized to the point at which Egyptian fractions enter as a limiting factor. This article also hashes out many of the details glossed over in this post.