Diophantine sets and the integers

Standard
Hilbert's problems

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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…