Hhhhmm. Ok. So I invented this solution in 2009 at what you might call a "peak mental moment", by a pool in Palm Springs, CA, after about 6 hours of writing on napkins. I'm not a mathematician. I don't think I'm even a great programmer, since there are probably much better ways of solving the thing I was trying to solve. And also, I'm not sure how I even came up with the reduction; I probably was wrong or made a typo (missing the +1?), and I'm not even certain how I could come up with it again.

2^x & 2^y ...is the & a bitwise operator...???? That would produce a unique ID? That would be very interesting, is that provable?

Primes take too much time.

The thing I was trying to solve was: I had written a bitcoin poker site from scratch, and I wanted to determine whether any players were colluding with each other. There were too many combinations of players on tables to analyze all their hands versus each other rapidly, so I needed to write a nightly cron job that collated their betting patterns 1 vs 1, 1 vs 2, 1 vs 3... any time 2 or 3 or 4 players were at the same table, I wanted to have a unique signature for that combination of players, regardless of which order they sat in at the table or which order they played their hands in. All the data for each player's action was in a SQL table of hand histories, indexed by playerID and tableID, with all the other playerIDs in the hand in a separate table. At the time, at least, I needed a faster way to query that data so that I could get a unique id from a set of playerIDs that would pull just the data from this massive table where all the same players were in a hand, without having to check the primary playerID column for each one. That was the motivation behind it.

It did work. I'm glad you were curious. I think I kept it as the original algorithm, not the reduced version. But I was much smarter 15 years ago... I haven't had an epiphany like that in awhile (mostly have not needed to, unfortunately).