Talk:Wolstenholme prime

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General Comments. I have an undergrad degree in Mathematics, but not much beyond that. :)
Generally a very good article. The section on 'Wolstenholme primes and irregular primes' needs to be brought up to the standard of the rest of the article, with the info at the end being turned into references. Unfortunately, papers yet to be published can not be used in the article, however, for an article like this, mentioning them on the talk page as a reminder would be just fine. Also, the concept of a line that says "for a proof see article blah-blah" should probably be removed. I think the statement previous should just have the reference for article blah-blah. Something that may be a concern is that it there *may* not be enough for two articles between the Wolstenholme prime and the Wolstenholme's Theorem, but I'd put this one out there and see if anyone proposes merging them at a later date. Given that this article is more complete than the one on Wolstenholme's Theorem, I doubt that anything would be lost from this article in any possible future merge. Again, I wish all articles that were started on Wikipedia were at this level of quality.Naraht (talk) 12:16, 17 March 2011 (UTC)

The two papers yet to be published are the following

• Luca, F.; Young, P. T., "Some p-adic congruences for p^q-Catalan numbers" (PDF), Applications of Fibonacci Numbers, ?
• Buhler, J. P.; Harvey, D., "Irregular Primes to 163 Million", Mathematics of Computation, ?, doi:10.1090/S0025-5718-2011-02461-0 {{citation}}: Unknown parameter |arXiv= ignored (|arxiv= suggested) (help)

Accidental deletion attempt

I previously blanked the page. I accidentally clicked on the Wikilink in my userspace, which after the move obviously redirects here, because I tried to delete the (non existing) page in my userspace. I undid this edit. Yeah and I know that the move of a page in fact renames it, I simply didn't think. Cheers. Toshio Yamaguchi (talk) 20:19, 7 April 2011 (UTC)

No harm, no foul. At *worst* you would have an administrator coming to the page and putting a note on your page on the inappropriateness of using the tag, whereupon those of us involved in the move would have put nice pretty things on your talk page to make up for it. :) Naraht (talk) 20:25, 7 April 2011 (UTC)

Toshio, do you want the link in your userspace deleted? EdJohnston (talk) 22:18, 7 April 2011 (UTC)

I already deleted it. But thanks for asking, I appreciate it. Toshio Yamaguchi (talk) 22:24, 7 April 2011 (UTC)

Actually, the link is still there. EdJohnston (talk) 22:29, 7 April 2011 (UTC)

Ah, right, the redirect still exists. Yes, it can be deleted. Thanks. Toshio Yamaguchi (talk) 23:13, 7 April 2011 (UTC)

Done. EdJohnston (talk) 23:28, 7 April 2011 (UTC)

Table listing searches for Wolstenholme primes

I would like to hear the opinions of other editors regarding what to do with the table in 'Search and current status'. In some sense, only the last three searches, namely the two searches performed by McIntosh and the search by Trevisan & Weber were actually searches for Wolstenholme primes, although McIntosh states in his 1995 paper about the two known Wolstenholme primes, that "The first was found (though not explicitly reported) by Selfridge and Pollak (Notices Amer. Math. Soc. 11 (1964), 97), and later confirmed by W. Johnson and S. S. Wagstaff" and "The second was found by J. Buhler, R. Crandall, R. Ernvall, and T. Metsänkylä, and later, independently, by the author."). I think this is not true. Neither selfridge & Pollack, nor Johnson or Ernvall & Metsänkylä did actually search for Wolstenholme primes. They all searched for irregular primes and Johnson simply discovered that p = 16843 has the property that (p, p - 3) is an irregular pair for this prime. It seems he was not aware of the fact that this means p is a Wolstenholme prime. Likewise, Buhler, Crandall, Ernvall & Metsänkylä discovered that the pair (p, p - 3) is irregular for p = 2124679. The term Wolstenholme prime was first introduced by McIntosh in his 1995 paper. On the other hand, following properties are all equivalent:

• p satisfies :${2p-1 \choose p-1}\equiv 1{\pmod {p^{4}}},$
  
• p divides the numerator of the Bernoulli number B_p−3
  
• (p, p - 3) is an irregular pair
  

I am not sure if there are published proofs for the equivalence of the three definitions above, but if there were, then this would mean every search for irregular primes is also a search for Wolstenholme primes. This would also mean, that the sections Irregular pairs and History previously moved to the article Regular prime from here should probably be reinserted into this article. I think, any search for irregular primes will necessarily identify any Wolstenholme primes in a search interval. Toshio Yamaguchi (talk) 18:34, 9 May 2011 (UTC)

The search for irregular pairs is more general than the search for Wolstenholme primes, and I don't see any reason why the sections Irregular pairs and History from the Regular prime article should be moved here. Instead, this issue should clarified and the reader should be referred to the Regular prime article for further details. Maxal (talk) 20:42, 10 May 2011 (UTC)

I think you are right. For example if someone searched for irregular primes in some range above lets say 10⁹ and discovered (hypothetically) that for p=1000000007 (p, p - 3) was irregular and published this, claiming it would be a Wolstenholme prime on Wikipedia without such a statement in the publication would be original research, even if it were true. Thanks. I think I will remove the table in the section 'Search and current status' from the article and add a description of the three searches for Wolstenholme primes (the two by McIntosh and that by Trevisan & Weber) to the section using prose. Toshio Yamaguchi (talk) 15:44, 12 May 2011 (UTC)