Change of Lead in Prime Number Race 4n+1 vs. 4n-1

Theorem

Consider the prime number race between $4 n + 1$ and $4 n - 1$.

While the prime numbers of the form $4 n - 1$ appear usually to be in the majority, the lead changes from one to the other an infinite number of times.

Proof

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Historical Note

According to David Wells in his $1997$ work Curious and Interesting Numbers, 2nd ed., this result was demonstrated by John Edensor Littlewood.

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): $26,861$