# The Erdős–Straus conjecture

I came across this as a result of links to stories about Hilbert’s tenth problem, and it looks fun, so I thought I’d write a little about it.

The Erdős–Straus conjecture is that for any integer $n \geqslant 4$ then $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ where $x$, $y$, and $z$ are positive integers.

This is equivalent to a diophantine equation:

$xyn^1 + xzn^1 + yzn^1 - 4xyz = 0$

Which is, apparently, trivially solvable for composite (non-prime) numbers. And we can obviously see that if $n=pq$ then if we had an expansion $\epsilon$ for $\frac{4}{p}$ then the expansion for $\frac{4}{n}$ would simply be $\frac{\epsilon}{q}$ – so if we found the conjecture true for a prime then it would be true for all that prime’s multiples. Hence to disprove the conjecture one needs to find a prime for which it is false.

And, indeed, the truth of the conjecture is an open question, though computationally it has been verified as true up to $10^{14}$.