cartesian product … stuff about computing mostly

Diophantine sets and the integers

Hilbert's problems — 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).

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…

Hilbert’s 10.5th Problem (rjlipton.wordpress.com)
On the number of solutions to 4/p = 1/n_1 + 1/n_2 + 1/n_3 (terrytao.wordpress.com)
The Omega Man (beatsnpeace.wordpress.com)
Counting the number of solutions to the Erdös-Straus equation on unit fractions (terrytao.wordpress.com)
Hilbert’s seventh problem, and powers of 2 and 3 (terrytao.wordpress.com)

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Adrian McMenamin

September 20, 2011

Diophantine equation, Hilbert’s problems, Hilbert’s tenth problem