Tagged: Hilbert’s tenth problem

September 22, 2011

The Erdős–Straus conjecture

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 $10^{14}$ .

Diophantine sets and the integers (cartesianproduct.wordpress.com)

September 20, 2011

Diophantine sets and the integers

Image via Wikipedia

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).