# The Erdős–Straus conjecture

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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 $10^{14}$.

# Diophantine sets and the integers

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This is not some great revelation, but it interested me, so might interest some readers (I got it from The Honors Class: Hilbert’s Problems and Their Solvers).

A diophantine equation is one of the form $ax^4+bx^3+cx^2+dx+e=0$ and a diophantine set is a set of numbers to solve a diophantine equation (Hilbert’s 10th problem was to find an algorithm to solve these in the general case – a task we now know to be impossible).

One diophantine set is the integers – which are a solution to $x=a^2+b^2+c^2+d^2$

Eg.,

$0=0^2$
$1=1^2$
$2=1^2+1^2$
$3=1^2+1^2+1^2$
$4=2^2$
$5=2^2+1^2$
$6=2^2+1^2+1^2$
$7=2^2+1^2+1^2+1^2$

$15=3^2+2^2+1^2+1^2$

$24=4^2+2^2+2^2$

and so on…