The Erdős–Straus conjecture

An Erdős-Diophantine graph with five points.
Image via Wikipedia

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} .