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

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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:

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

This sequence is A007351 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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


Proof


Sources