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The Erdos-Straus Conjecture says that, for all $n > 1$, there exist positive integers $x,y,z$, such that $\dfrac{4}{n} = \dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}$. A generalization due to Schinzel postulates that $\dfrac{k}{n} = \dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}$ has a solution for all fixed $k$ and large enough $n$. Two questions:

As far as I know it hasn't been proved that the set of $n$ that doesn't satisfy the E-S Conjecture is less than, say, $n^{0.99}$. Is something like this known assuming some well-known conjecture (other than E-S itself of course!)?

Assuming Schinzel's generalization, we could ask: What is the largest $n = n(k)$ without a solution? Is there a conjecture/heuristic as to how large this $n$ should be? I believe it can be shown that this $n$ grows larger than $k^{8/7}$.

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Quite recent status and results are reported in the paper by Christian Elsholtz and Terence Tao, "Counting the number of solutions to the Erdos-Straus equation on unit fractions," arXiv:1107.1010v3 [math.NT]. – Joseph O'Rourke Nov 11 '11 at 16:15

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