52/15 fixed gear ratio - what should I change to?
Different from the other answer, I read the question as OP wanting to change to lower gear ratio without changing chain length. 52/15 is an extremely high ratio.

For all practical purposes, if you want to keep the chain length same, the sum of numbers of teeth in cog and ring should stay the same. 

The changing geometry does affect the result a bit, but the difference it will make is less than a half link's length. The exact math is a bit hairy, since you have to take into account that the part of chain that wraps around cog is not actually part of circle, but N-gon that has its angles at chain pins.

My recommendation would be to bite the bullet and get a chain tool and not try to optimize cog sizes. You will need one anyway later, and if the old cogs and chain were worn, it's best to replace the chain at the same time. A normal sized chainring will give you more options to play with ratios later, and prime numbers 41, 43 and 47 would let you use any cog without thinking about skid patches.

With lower ratio you might need to take the skid points into account.