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 and be any given natural numbers and be a given prime. Let be the unique number not divisible by and smaller than for which a exists such that is the closest underapproximation of possible**. Let be the least common multiple of the first natural numbers and let be such that . Then: divides if, and only if, divides .

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

Let be the number of digits of the denominator of the -th harmonic number, written in base . Then: converges to .

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 , and , we have

(). So our theorem says iff , which is not the case.

If , and , we have

(). So our theorem says iff , which is not the case.

If , and , we have (). So our theorem says iff , which is the case. In general we have (assuming , otherwise would just equal and our theorem then trivializes):

,

and

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