## More on unit fractions, greatest common divisors, least common multiples and such

Just a quick update, I’ll elaborate later. The technique I used 2 posts ago, in proving that $g_n > 1$ infinitely often, can be quite easily generalised and yield a lot more that way. And it also gives some insight in why it’s hard to show $g_n = 1$ infinitely often; maybe it’s not true! The theorem I will prove later (either today or tomorrow) will be:

Let $n$ be a given natural number and $p$ be a given prime. Let $a \in \aleph$ be the unique number smaller than $p$ such that $ap^k \le n < (a+1)p^k$ for some $k \in \aleph_0$. Then the following two are equivalent:

1) $p | X_n$
2) $p | X_a$

Where $\dfrac{X_n}{L_n}$ $= \sum_{i=1}^n \dfrac{1}{i}$, with $L_n$ the least common multiple of $\{1,2,..,n\}$