Largest Known Lead by 4n+1 in Prime Number Race
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Theorem
In the prime number race between prime numbers of the form $4 n - 1$ and $4 n + 1$, the highest known stretch of integers where $4 n + 1$ is not less than $4 n - 1$ is between $18 \, 465 \, 126 \, 293$ and $19 \, 033 \, 524 \, 538$.
Proof
First note that:
\(\ds 18 \, 465 \, 126 \, 257\) | \(=\) | \(\ds 4 \times 4 \, 616 \, 281 \, 564 + 1\) | ||||||||||||
\(\ds 18 \, 465 \, 126 \, 293\) | \(=\) | \(\ds 4 \times 4 \, 616 \, 281 \, 573 + 1\) |
where it can be seen that prime numbers of the form $4 n + 1$ are locally increasing, and:
\(\ds 19 \, 033 \, 524 \, 533\) | \(=\) | \(\ds 4 \times 4 \, 758 \, 381 \, 133 + 1\) | ||||||||||||
\(\ds 19 \, 033 \, 524 \, 539\) | \(=\) | \(\ds 4 \times 4 \, 758 \, 381 \, 135 - 1\) |
where it can be seen at that point prime numbers of the form $4 n - 1$ may now be in front, but this is inconclusive.
This theorem requires a proof. In particular: I'm not actually sure this is correct. If $4 n - 1$ is now in front for the first time since $18 \, 465 \, 126 \, 293$, I would expect the previous prime also to be of the same form -- but it is not. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $18,465,126,293$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $18,465,126,293$