  We create a simple test for distinguishing between sets of primes and random
numbers using just the sum-of-digits function. We find that the
sum-of-the-digits of prime numbers does not have an equal probability of being
odd or even. The authors know of no reason why prime numbers should bias
themselves towards a particular parity in their sums of digits, but our
empirical tests show a very strong bias; strong enough that we are able to
devise a test to reliably differentiate between collections of prime numbers
versus random numbers by looking only at their sums of digits. We are also able
to create similar tests for products of primes. We are even able to test
"tainted" sets with mixtures of primes and random numbers: as the percentage of
(randomly chosen) prime numbers in a set of random numbers is varied, we get a
reliable, linear change in our parity measure. For example, when we add up the
digits of prime numbers in base 10, their sum is significantly more likely to
be odd than even. This effect persists across base changes, although which
parity is more common might change. Note that the last digit being odd in base
10 simply reverses the parity. We have tested this for the first fifty million
primes -- not primes up to 50,000,000, but the first 50,000,000 prime numbers
-- and have found that this effect persists, and does so in a predictable
manner. The effect is quite significant; for 50,000,000 primes in base 10, the
number of primes which have an odd sum-of-digits is about an order of magnitude
farther away from the mean than expected. We have run multiple tests to try and
understand the source of this bias, including investigating primes modulo
random numbers and adjusting for Chebyshev's bias. None of these tests yielded
any satisfactory explanation for this phenomenon.
