## Two (happy) notes on unit fractions

After some encouragement of Ernie Croot (yes, the man that proved the Erdös-Grahamproblem), I managed to find the following generalization of an earlier shown theorem;

Let $n$ and $m$ be any given natural numbers and $p$ be a given prime. Let $a \in \aleph$ be the unique number not divisible by $p$ and smaller than $p^m$ for which a $k \in \aleph_0$ exists such that $ap^k$ is the closest underapproximation of $n$ possible**. Let $L_n$ be the least common multiple of the first $n$ natural numbers and let $X_n$ be such that $\dfrac{X_n}{L_n} = \displaystyle \sum_{i=1}^n \dfrac{1}{i}$. Then: $p^m$ divides $X_n$ if, and only if, $p^m$ divides $X_a$.

On another happy note, on Wolfram Mathworld I found the following conjecture (slightly generalized by myself):

Let $D_n$ be the number of digits of the denominator of the $n$-th harmonic number, written in base $b$. Then: $\dfrac{n}{D_n}$ converges to $\log{b}$.

And I think the best application of everything I have been able to show so far concerning the problems of Erdös and Graham, is an easy proof of the above (sweet!) conjecture 🙂

** Since this definition might be vague, here are some examples:
If $n = 100$, $p = 7$ and $m = 1$, we have $a = 2$
($2*7^2 = 98 \le 100$). So our theorem says $7 | X_{100}$ iff $7 | X_2$, which is not the case.
If $n = 100$, $p = 7$ and $m = 3$, we have $a = 100$
($100*7^0 = 100 \le 100$). So our theorem says $7^3 | X_{100}$ iff $7^3 | X_{100}$, which is not the case.
If $n = 100$, $p =5$ and $m = 2$, we have $a = 4$ ($4*5^2 = 100 \le 100$). So our theorem says $5^2 | X_{100}$ iff $5^2 | X_4$, which is the case. In general we have (assuming $n > p^m$, otherwise $a$ would just equal $n$ and our theorem then trivializes):
$k = \left\lfloor\log_p{n}\right\rfloor - m + 1$,
$a' = \left\lfloor\dfrac{n}{p^k}\right\rfloor$ and
$a = \dfrac{a'}{(a', p^m)}$