## Lower bounds on Erdos-Strauss and relatives

Let $k$ be given positive integer and let $n = n_k$ be the largest integer, such that $\dfrac{k}{n}$ cannot be written as a sum of $3$ unit fractions, that is $\dfrac{k}{n} = \dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}$ is not solvable in positive integers $x, y, z$. If all is well, $n$ is finite for every $k$, although even showing that for $k = 4$, the first non-trivial case, would essentially solve a 60-year old conjecture!

1) Give a superlineair bound on $n$
2) Give a bound on $n$ that grows asymptotically faster than $k^{8/7}$

Beeteedubs, if you don’t mind, please don’t solve E-S, or at least share your thoughts. I wanna be a part of it like really really badly 🙂