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

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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


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