An Euler brick satisfies the Diophantine Equations:
$a^2+b^2=d^2$
$a^2+c^2=e^2$
$b^2+c^2=f^2$
Where a,b,c,d,e, and f are integers.
Has anyone proved the solutions are recursively enumerable?
Or does that fact that there are infinite solutions mean the set is not recursively enumerable?
Let $S\subseteq\Bbb N^6$ be the set of all sextuples $(a,b,c,d,e,f)$ satisfying $a^2+b^2=d^2,a^2+c^2=e^2,b^2+c^2=f^2$. To show that this is a recursively enumerable set, we need to specify an algorithm which, given $a,b,c,d,e,f$ as an input, stops iff $(a,b,c,d,e,f)\in S$. Consider the following algorithm:
if(a^2+b^2=d^2 AND a^2+c^2=e^2 AND b^2+c^2=f^2) STOP else LOOP FOREVER(loop forever can be achieved e.g. by making the program execute one command without end)
It is clear that this algorithm will only stop if the tuple is in $S$. Hence we have shown that $S$ is recursively enumerable.