Why is the size 127 (prime) better than 128 for a hash-table?

Question

Supposing simple uniform hashing, that being, any given value is equally like to hash into any of the slots of the hash. Why is it better to use a table of size 127 and not 128?

Answer 1

I cannot prove it anymore, although I remember having to do so in an exam at university a million years ago, but optimal hash sizes are not merely prime. You want to pick a prime number N such that N = 4*M − 1 (where M is also an integer).

That makes 31 a better number of buckets than 29. M is 8 when N is 31, but there is no integral M when N is 29.

As I said, I no longer remember the math to prove this. It was in a theory course taught by Rachel Manber, Udi’s wife, about 25 years ago or so.