Largest Known Lead by 4n+1 in Prime Number Race

From ProofWiki
Jump to navigation Jump to search

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

Retrieved from "https://proofwiki.org/w/index.php?title=Largest_Known_Lead_by_4n%2B1_in_Prime_Number_Race&oldid=384986"