Jones' conjecture on binomial congruences
For , let
A prime satisfying is considered in the context of Wolstenholme's theorem. Jones' conjecture.
This asserts that the known congruence from Wolstenholme's theorem has no other solutions. The source gives no resolution, so the conjecture remains open.
References
Primary source
Saud Hussein, “New conjecture related to a conjecture of McIntosh”, arXiv:1802.01909 (2019).
Progress summary
No complete proof or counterexample has been found publicly; only finite checks and partial exclusions are known.
Jones' conjecture asserts that the congruence for the central binomial coefficient occurs exactly at primes at least . The converse of Wolstenholme's theorem remains unresolved in the available literature.
Known results
- Trevisan and Weber, 2001: no even positive integer satisfies the corresponding congruence.
- A 2018 paper reports validity for prime powers up to and computational verification for all integers up to .
- The same paper proves further infinite families of nonsolutions, including certain integers of the form .
Current status (as of September 2026): The conjecture remains open; even integers and several other families are excluded, and extensive finite computations support it, but no complete proof or counterexample is recorded.
Sources
- ar5iv.labs.arxiv.org
- ar5iv.labs.arxiv.org
- arxiv.org
- arxiv.org
- researchgate.net
- maths.nju.edu.cn
- mdpi.com
- semanticscholar.org
- math.ecnu.edu.cn
- en.wikipedia.org
- mathoverflow.net
- ar5iv.labs.arxiv.org
- ar5iv.labs.arxiv.org
- ar5iv.labs.arxiv.org
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
Solutions 0
No solutions have been posted yet.