If you haven’t read everything I wrote last week, but would still like to solve an Erdös-Graham-problem or two with me, here is, basically, everything I know so far:

Let n be a given natural number and p be a prime smaller than n. Let a \in \aleph be the unique number smaller than p such that ap^k \le n < (a+1)p^k holds for some k \in \aleph_0. Let L_n be the least common multiple of \{1,2,..,n\} and X_n be such that \dfrac{X_n}{L_n} = \displaystyle \sum_{i=1}^n \dfrac{1}{i}. And, at last, let u_{y,z} and v_{y,z} be coprime positive integers, such that, for y,z \in \aleph\displaystyle \sum_{i=y}^z \dfrac{1}{i} = \dfrac{u_{y,z}}{v_{y,z}}. Then:

\text{ }

1) p | X_n,  p | X_a and p | X_{p-1-a} are all equivalent

2) If p > (1 + o(1))\dfrac{n}{\log{n}}, then p does not divide X_n

3) If n is not a prime power, X_n \neq X_{n-1} \pmod{p} implies n = ap^k

4) For every fixed y, there is a z \le 6y, for which v_{y,z} < v_{y,z-1}

\text{ }

Btw, one of the problems is: prove or disprove that the greatest common divisor of X_n and L_n is equal to 1 for infinitely many n. Another one is to find the smallest (or at least a better upper bound for) z = z(y), such that v_{y,z} < v_{y,z-1}.


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