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Feb 2, 2010 at 17:34 comment added Reid Barton @B.: The current answers are just as relevant now as before your edit.
Feb 2, 2010 at 16:49 comment added B. Bischof @fedja Uniqueness is not necessary, I had made the mistake of assuming S was a set. In reality it is a multiset. Thank you for your relevant comment, I have corrected the question and added an example of the situation which you pointed out. Also, I added some details on the current method he is using to decompose S.
Feb 2, 2010 at 16:46 history edited B. Bischof CC BY-SA 2.5
changed to multiset, added example, added current algorithm
Feb 2, 2010 at 6:27 answer added Bjorn Poonen timeline score: 11
Feb 2, 2010 at 6:04 answer added Qiaochu Yuan timeline score: 4
Feb 2, 2010 at 5:48 comment added fedja It is unclear whether you require the uniqueness of the representation of every number in $S$ in the form $a+b$. You do not in the list of properties but your remark about cardinalities strongly suggests that you do. Can you clarify this?
Feb 2, 2010 at 4:11 history asked B. Bischof CC BY-SA 2.5