# Definition:Prime Number Race

## Definition

The set of prime numbers $\Bbb P$ may be partitioned into subsets according to a particular property.

A prime number race is a comparison of the count of the number of prime numbers in each partition with increasing $p \in \Bbb P$.

## Examples

### $4 n + 1$ vs. $4 n - 1$

The sequence of prime numbers at which the prime number race between prime numbers of the form $4 n - 1$ and $4 n + 1$ are tied begins:

$2, 5, 17, 41, 461, 26 \, 833, 26 \, 849, 26 \, 863, 26 \, 881, 26 \, 893, 26 \, 921, 616 \, 769, \ldots$

### $3 n + 1$ vs. $3 n - 1$

In the prime number race between prime numbers of the form $3 n - 1$ and $3 n + 1$, the prime numbers of the form $3 n + 1$ overtake those of the form $3 n - 1$ for the first time at $608 \, 981 \, 813 \, 029$.