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 🙂

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