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

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## Contents

## 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

- 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): $26,861$