# A Peculiarity in the Parity of Primes

## TL;DR

This paper uncovers a strong bias in the sum-of-digits parity of prime numbers, enabling a reliable test to distinguish primes from random numbers, with persistent effects across large datasets and various bases.

## Contribution

The authors reveal a novel parity bias in prime sums of digits and develop a simple, effective test to differentiate primes from random numbers based solely on this property.

## Key findings

- Prime sums of digits are significantly biased towards odd parity.
- The bias persists across different bases and large datasets.
- The bias can be used to reliably distinguish primes from random numbers.

## Abstract

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.