# Diophantine sets and the integers

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…