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\}

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