## Clearing the PEN-Book

Just a quick post to say ‘Wieeeee’; today I solved the following problems, all from PEN;

1) Prove that for every n, the set of unit fractions contains a maximal arithmetic progression of length n. Maximal means that, if $\displaystyle \frac{p}{q}$ is the common difference, $\displaystyle \frac{1}{x_1}$ is the largest and $\displaystyle \frac{1}{x_n}$ is smallest term of the arithmetic progression, then both $\displaystyle \frac{1}{x_1} + \frac{p}{q}$ and $\displaystyle \frac{1}{x_n} - \frac{p}{q}$ are not unit fractions

2) Prove that a 3*3 magical square cannot contain only Fibonacci numbers

3) Prove that for every n, there are 2 consecutive integers that are the product of n integers

Since my thought process, nor my proofs are interesting, I won’t give any details here. Although I must say that changing ‘3*3’ to ‘m*m’ in problem 2) and disallowing the use of induction in problem 3) makes those problems slightly more interesting. Even better: disallowing the use of dirichlet’s theorem on arithmetic progressions in problem 1). I have, so far, not been able to solve that one, although I didn’t think that hard about it yet and I might be indoctrinated, of course. I also tried to prove that there are 2 consecutive integers that are the product of n distinct primes for every n, which, to me, is a much more natural question to ask than problem 3). But I quickly realised that there is a reason that problem 3) is what it is, because my generalization trivially implies that there are infinitely many consecutive numbers with the same number of divisors. And this is known to be a hard problem and was, according to Wikipedia, only settled in 1984, so I just gave up on that. Maybe the following is ‘reasonably’ attackable: prove that there are 2 consecutive integers that are both the product of n distinct prime powers for every n. But I wasn’t able to solve that one either. If you have any interesting thoughts on anything I said here, don’t hesitate to comment. Anyway, wieeeee