## Generalizing generalities

Theorem.
Let $L_n$ be the least common multiple of $\{1,2,..,n\}$ and $X_n = L_n \displaystyle \sum_{i=1}^n \dfrac{r_i}{i s_i}$ with $r_i, s_i \in Z$ for all $i$, $r_{i+m} = r_i$ for some $m$ and $s_{i+m'} = s_i$ for some $m'$. Then $\gcd(X_n, L_n) > 1$ for infinitely many $n$.

Proof.
It is clear that it is necessary that the numerator of $X_n$ does not equal $1$ infinitely often. So let’s start with that. Assume that the numerator of $X_{n-1}$ does equal $1$ for some large $n$, where $n$ is not a prime power. Then: $|X_n| + |X_{n-1}| \ge |X_n - X_{n-1}| = \left|\dfrac{L_n r_n}{n s_n}\right| > \dfrac{L_n}{n^2} > 2$. And since $|X_{n-1}| \le 1$, we have $|X_n| > 1$, which is never a unit fraction. Let $a$ be an integer for which $a \ge \max(m, m', \displaystyle \max_i (|s_i|))$ and $p$ be a prime larger than $a$ such that $p$ divides $X_a$. Note that if such a prime doesn’t exist for any $a$, we would have that all the prime divisors of $X_a$ (which, as we saw, exist for infinitely many $a$), are smaller than or equal to $a$. But a prime divisor of $X_a$ which is smaller than or equal to $a$ also divides $L_a$. And this would immediately imply our theorem. So we may assume the existence of a prime $p$ $> a \ge$ $\max(m, m', \displaystyle \max_i(|s_i|))$ dividing $X_a$. Now, let $k$ be any (positive) multiple of lcm$(\varphi(m), \varphi(m'))$ and set $n = ap^k$;

$X_n = L_n \displaystyle \sum_{i=1}^n \dfrac{r_i}{i s_i}$
$\equiv L_n \displaystyle \sum_{i=1}^a \dfrac{r_{ip^k}}{ip^k s_{ip^k}}$ $\pmod{p}$
$\equiv \dfrac{L_n}{p^k} \displaystyle \sum_{i=1}^a \dfrac{r_i}{i s_i}$      $\pmod{p}$
$\equiv 0$                      $\pmod{p}$