# Prime Number Race/Examples/4n+1 vs. 4n-1

## Example of Prime Number Race

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:

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

The details of this prime number race is as follows:

 $\displaystyle p = 2$ $:$ $\displaystyle$ Both are equal $\displaystyle 2 < p < 5$ $:$ $\displaystyle$ $4 n - 1$ leads $\displaystyle p = 5$ $:$ $\displaystyle$ Both are equal $\displaystyle 5 < p < 17$ $:$ $\displaystyle$ $4 n - 1$ leads $\displaystyle p = 17$ $:$ $\displaystyle$ Both are equal $\displaystyle 17 < p < 41$ $:$ $\displaystyle$ $4 n - 1$ leads $\displaystyle p = 41$ $:$ $\displaystyle$ Both are equal $\displaystyle 41 < p < 461$ $:$ $\displaystyle$ $4 n - 1$ leads $\displaystyle p = 461$ $:$ $\displaystyle$ Both are equal $\displaystyle 461 < p < 26 \, 833$ $:$ $\displaystyle$ $4 n - 1$ leads $\displaystyle p = 26 \, 833$ $:$ $\displaystyle$ Both are equal $\displaystyle 26 \, 833 < p < 26 \, 849$ $:$ $\displaystyle$ $4 n - 1$ leads $\displaystyle p = 26 \, 849$ $:$ $\displaystyle$ Both are equal $\displaystyle 26 \, 849 < p < 26 \, 863$ $:$ $\displaystyle$ $4 n + 1$ leads, for the first time $\displaystyle p = 26 \, 863$ $:$ $\displaystyle$ Both are equal $\displaystyle 26 \, 863 < p < 26 \, 881$ $:$ $\displaystyle$ $4 n - 1$ leads $\displaystyle p = 26 \, 881$ $:$ $\displaystyle$ Both are equal $\displaystyle 26 \, 881 < p < 26 \, 893$ $:$ $\displaystyle$ $4 n - 1$ leads $\displaystyle p = 26 \, 893$ $:$ $\displaystyle$ Both are equal $\displaystyle 26 \, 893 < p < 26 \, 921$ $:$ $\displaystyle$ $4 n - 1$ leads $\displaystyle p = 26 \, 921$ $:$ $\displaystyle$ Both are equal $\displaystyle 26 \, 921 < p < 616 \, 769$ $:$ $\displaystyle$ $4 n - 1$ leads