TITLE: Can two positive integers be uniquely recovered from their difference and XOR?
QUESTION [6 upvotes]: As part of an answer to a Stack Overflow question I made the assumption that if I choose two distinct positive integers $m$ and $n$, them give you $m - n$ and $m$ XOR $n$, then you can uniquely determine what $m$ and $n$ were.  For all the examples I've tried this seems to work correctly, though I have no reason to believe that this should work in general.  Moreover, I'm not familiar enough with the interactions of differences (or sums, for that matter) and XOR to deivse a proof or counterexample.
Is my claim true?  If so, how would you go about proving it?  If not, is there a nice counterexample?
Thanks so much!

REPLY [9 votes]: I believe this is false.
Let $2^r \gt m \gt n$.
Then
$2^r + m$ and $2^r +n$ have same difference and XOR as $m,n$.

REPLY [2 votes]: How about (2,3) and (16,17)? plus some characters