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:

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

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:

\(\ds p = 2\)

\(:\)

\(\ds \)

Both are equal

\(\ds 2 < p < 5\)

\(:\)

\(\ds \)

$4 n - 1$ leads

\(\ds p = 5\)

\(:\)

\(\ds \)

Both are equal

\(\ds 5 < p < 17\)

\(:\)

\(\ds \)

$4 n - 1$ leads

\(\ds p = 17\)

\(:\)

\(\ds \)

Both are equal

\(\ds 17 < p < 41\)

\(:\)

\(\ds \)

$4 n - 1$ leads

\(\ds p = 41\)

\(:\)

\(\ds \)

Both are equal

\(\ds 41 < p < 461\)

\(:\)

\(\ds \)

$4 n - 1$ leads

\(\ds p = 461\)

\(:\)

\(\ds \)

Both are equal

\(\ds 461 < p < 26 \, 833\)

\(:\)

\(\ds \)

$4 n - 1$ leads

\(\ds p = 26 \, 833\)

\(:\)

\(\ds \)

Both are equal

\(\ds 26 \, 833 < p < 26 \, 849\)

\(:\)

\(\ds \)

$4 n - 1$ leads

\(\ds p = 26 \, 849\)

\(:\)

\(\ds \)

Both are equal

\(\ds 26 \, 849 < p < 26 \, 863\)

\(:\)

\(\ds \)

$4 n + 1$ leads, for the first time

\(\ds p = 26 \, 863\)

\(:\)

\(\ds \)

Both are equal

\(\ds 26 \, 863 < p < 26 \, 881\)

\(:\)

\(\ds \)

$4 n - 1$ leads

\(\ds p = 26 \, 881\)

\(:\)

\(\ds \)

Both are equal

\(\ds 26 \, 881 < p < 26 \, 893\)

\(:\)

\(\ds \)

$4 n - 1$ leads

\(\ds p = 26 \, 893\)

\(:\)

\(\ds \)

Both are equal

\(\ds 26 \, 893 < p < 26 \, 921\)

\(:\)

\(\ds \)

$4 n - 1$ leads

\(\ds p = 26 \, 921\)

\(:\)

\(\ds \)

Both are equal

\(\ds 26 \, 921 < p < 616 \, 769\)

\(:\)

\(\ds \)

$4 n - 1$ leads

Proof

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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: pp. 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): $26,861$