# Unique sum for all natural numbers

• Nov 3rd 2010, 11:41 AM
doug
Unique sum for all natural numbers
I am looking for two infinite subsets of the natural numbers ( $A, B \subseteqq N$), that any n natural numbers can be written in a unique way as a sum of an elemenet from each sets: n=a+b (where $a \in A, b \in B$.
Any help would be appreciated!
• Nov 3rd 2010, 01:34 PM
Opalg
Quote:

Originally Posted by doug
I am looking for two infinite subsets of the natural numbers ( $A, B \subseteqq N$), that any n (natural number) can be written in a unique way as a sum of an element from each sets: n=a+b (where $a \in A, b \in B$).

You could for example take A to be the set of all natural numbers whose decimal expression has a 0 in all the even-numbered places (counting from the right), and B to be the set of all natural numbers whose decimal expression has a 0 in all the odd-numbered places.

So for example the number n = 31416265 can be written 1010205 + 30406060 (with the first of those numbers being in A and the second one being in B).