Generalizing generalities

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.

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}


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