# 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

- Jan. 1978: Carter Bays and Richard H. Hudson:
*On the Fluctuations of Littlewood for Primes of the Form $4n \pm 1$*(*Math. Comp.***Vol. 32**,*no. 141*: 281 – 286) www.jstor.org/stable/2006277

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $26,861$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $461$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $26,861$