Talk:Wieferich prime

message: ===Added section===
Wieferich primes and Fermat numbers.
:A Fermat prime cannot also be a Wieferich prime.
* W.Banks,F. Luca and I. Shparlinski, “Estimates for Wieferich numbers,” Ramanujan J. 14. (2007), no. 3, 361–378.
===Commentary===
This line has been added frequently, with the reference added to the "further reading" section.  This is 
#Badly written
#Not written with a < ref > tag or  template.
#Not notable.
#Sourced, if at all, to that reference, which may, in turn, not be reliable, notable, or contain the result.
— Arthur Rubin  (talk) 19:28, 26 April 2008 (UTC)
thread_title:  Fermat numbers 

message: This line has been added frequently, with the reference added to the "further reading" section.  This is 
#Badly written
#Not written with a < ref > tag or  template.
#Not notable.
#Sourced, if at all, to that reference, which may, in turn, not be reliable, notable, or contain the result.
— Arthur Rubin  (talk) 19:28, 26 April 2008 (UTC)
thread_title: Commentary

message: Surely those were References not Further Reading?  Is the Silverman reference used in the text?  Richard Pinch (talk) 07:05, 2 August 2008 (UTC)
replies: text: This section is from 2008. We make new posts at the bottom but I happened to see your edit on my watchlist. The error may be temporary but I have added  with a url in  so now there is a working link to an archive. PrimeHunter (talk) 01:41, 14 October 2014 (UTC)
replies: text: I replaced the link with a link to an archived version of the site in the Wayback Machine. If the site comes back again, a direct link to the site can be added again. -- Toshio Yamaguchi 07:24, 14 October 2014 (UTC)
thread_title:  References 

message: Is the W. Meissner who found the Wieferich prime 1093 really the physicist Walter Meissner?  Richard Pinch (talk) 07:06, 2 August 2008 (UTC)
replies: text: I believe this is not the case, and that it was Waldemar Meissner who found the Wieferich prime (, ). Waldemar Meissner has a very short entry on Czech wiki which basically states "information is missing" --Jvs.cz (talk) 15:08, 5 October 2008 (UTC)
replies: text: Thanks. I have changed the article to link Waldemar Meissner. PrimeHunter (talk) 19:03, 5 October 2008 (UTC)
thread_title:  Meissner 

message: At the beginning of the article, a Wieferich prime is defined as a prime p such that p2 divides 2p-1-1. This is correct but I think most sources use to define a Wieferich prime as a prime number satisfying the congruence 2p-1 ≡ 1 (mod p2) (see for example ).
Thus I propose changing the first sentence of the lead seaction
from
In number theory, a Wieferich prime is a prime number p such that p2 divides 2p−1−1; compare this with Fermat's little theorem, which states that every odd prime p divides 2p−1−1.
to
In number theory, a Wieferich prime is a prime p satisfying the congruence 2p-1 ≡ 1 (mod p2); compare this with Fermat's little theorem, which states that every odd prime satisfies the congruence 2p-1 ≡ 1 (mod p).
Toshio Yamaguchi (talk) 18:20, 3 July 2010 (UTC)
thread_title:  Definition of a Wieferich prime 

message: Maxal, I reverted your last edits, because you changed the book sources for the search by Carlisle, Crandall and Rodenkirch to point to the english editions of the books. Please note that I assume good faith on your part, but the english and german editions of the books seem to differ from each other. The Carlisle, Crandall and Rodenkirch search is only mentioned in the german editions of those books. Thus the english editions do not verify the given information, whereas the german editions do. Btw the books only mention this search, but do not give the values of the near-Wieferich primes found. I received these values from Mark Rodenkirch on request via email. The last near-Wieferich prime in the current table is one of those values and they are not published in the books or anywhere else. Toshio Yamaguchi (talk) 10:27, 29 March 2011 (UTC)
replies: text: Actually, you are correct about the Carlisle, Crandall and Rodenkirch search—it's mentioned only in German edition of "The little book of bigger primes" ("Die Welt der Primzahlen: Geheimnisse und Rekorde"). So, I'm retaining it. But for "My numbers, my friends: popular lectures on number theory", it is pointless to refer to the German edition. The corresponding pages are missing at Google/Books but that does not disqualify the book as a source of information. Moreover, this is English Wikipedia and English sources are preferable for it than sources in any other language. Partially reverting back. Maxal (talk) 16:41, 29 March 2011 (UTC)
thread_title:  Revert of most recent edits 

message: For a cyclotomic generalization of the Wieferich property:(np − 1)/(n − 1) divisible by q2, there are solutions like
*:(35 − 1)/(3 − 1) = 112
:and even with exponents higher than 2, like in
::(196 − 1)/(19 − 1)  ≡ 0 (mod 73).
I removed the above statement from the article, since I was unable to find a source mentioning this kind of generalization of the Wieferich property. It has been placed here for further discussion. If anyone is able to provide sources for this, it can be pasted back into the article from here with a source. Toshio Yamaguchi (talk) 18:15, 14 April 2011 (UTC)
replies: text: I can not find a reference either to a 'general wieferich prime'.  I made a post at dozensonline / mathematics / number theory / sevenites in September 2012 about this.  I have been studying this for many years without finding a general name, so i use 'sevenite' (after 7 in base 18), for the general 1/p distribution.  There seems to be a quite a number of wikipedia edits after 2015 when a user found that the binary sevenites are Wieferich, the trinary sevenites are Marianoff, the decimal sevenites are Shank's.  David Well's "curious and interesting numbers" mentions examples of five sevenites, but only the entry for 1093 carry Wieferich's name.  E. Dickson's "History of the Theory of Numbers" has an unnamed twelve-page chapter four on the subject, but mentions Wieferich only in a five-line paragraph on the nineth page.  Using the name Wieferich is a bad proposition, since the research stretches nearly 200 years before his paper, and his paper has nothing to do with the general divisibility of p^n | a^{p-1}-1. Wendy.krieger (talk) 10:14, 14 April 2018 (UTC)
thread_title:  Generalizations 

message: This source calls a prime number p satisfying 2p − 1 ≢ 1 (mod p2) a Wieferich prime, while it seems that most other sources agree that a Wieferich prime is a prime satisfying 2p − 1 ≡ 1 (mod p2).
Should there be a note about this in the article? Toshio Yamaguchi (talk) 18:22, 9 June 2011 (UTC)
replies: text: It is amazing that he even gives "examples" - he should have found out at least then, that he made something very wrong here... --Gotti 11:34, 14 June 2011 (UTC)
replies: text: Yeah, I think I will add the source to Further reading with a note pointing out this inconsistency. A reader searching Google books might stumble over this and after reading our article here and other sources, this might cause confusion to the reader. Toshio Yamaguchi (talk) 12:35, 14 June 2011 (UTC)
thread_title:  Opposing definitions of a Wieferich prime 

message: In the section about the connection with the Mersenne and Fermat primes it says
"A prime divisor p of Mq, where q is prime, is a Wieferich prime if and only if p2 divides Mq."
Furthermore it says
"Thus, a Mersenne prime cannot also be a Wieferich prime"
I do not see how the second statement follows from the first one. Unless someone can provide a source connecting these two statements in that way, I am going to remove the second statement, as this looks like original research. Toshio Yamaguchi (talk) 13:24, 29 November 2011 (UTC)
replies: text: A Mersenne prime is a prime divisor of itself. If Mq is prime then p=Mq is a prime divisor of Mq, but p2 = Mq2 does clearly not divide Mq. PrimeHunter (talk) 14:45, 29 November 2011 (UTC)
replies: text: Okay, I see it now. Thanks for the explanation. Toshio Yamaguchi (talk) 15:09, 29 November 2011 (UTC)
thread_title:  Connection with Mersenne primes 

message: Merging the contents from the List of near Wieferich primes article would benefit this article, by making it more complete, and hence, more encyclopedic. After a merger, the addition of a collapsed table would address the length of the finalized table. Northamerica1000(talk) 22:53, 28 December 2011 (UTC)
replies: text: I don't think the complete list should be in this article. A list of all near Wieferich primes in [1, 3] can be found in OEIS as . Toshio Yamaguchi (talk) 08:49, 29 December 2011 (UTC)
replies: text: This was exactly my argument. Merging does not make sense as all essential information is already present in Wieferich prime article. The article List of near Wieferich primes should be simply deleted. Maxal (talk) 01:10, 30 December 2011 (UTC)
replies: text: I nominated the list article for deletion (see Wikipedia:Articles for deletion/List of near Wieferich primes). Toshio Yamaguchi (talk) 01:28, 30 December 2011 (UTC)
thread_title:  Proposed merger 

message: In the section 'Connection with Mersenne and Fermat primes' I included the following statement:
It was observed that M1092 is divisible by 10932 and M3510 is divisible by 35112.
I did this, because http://www.elmath.org/index.php?id=display_subject&subject=2 says this is a "remarkable" fact. I don't have access to the paper by Guy which Miroslav Kures cites on the Wieferich@Home project homepage, but thinking about this again I believe this observation is rather trivial. Given that for a Wieferich prime qp(2) ≡ 0 (mod p) and the numerator of the Fermat quotient is always Mp-1 it follows as a corollary that for a Wieferich prime . Thus I believe the statement should perhaps be removed. Opinions? Toshio Yamaguchi (talk) 15:41, 8 January 2012 (UTC)
replies: text: I removed it. Toshio Yamaguchi (talk) 21:33, 9 January 2012 (UTC)
thread_title:  Is this property remarkable? 

message: I created a navbox template for the listing of unsolved problems (see User:Toshio Yamaguchi/Template:Unsolved 2) and propose to remove the current instance of Template:Unsolved at Wieferich prime#History and search status and replace it with this navbox (after having been moved into template namespace). See my sandbox for how the template looks like. Do other editors agree with this step? -- Toshio Yamaguchi (tlk−ctb) 21:51, 2 June 2012 (UTC)
replies: text: I disagree.  Navboxes are to be used for navigation, not for explanations per WP:NAVBOX.  I don't particularly like  either, but it has passed 4 deletion discussions already. Ryan Vesey Review me! 23:57, 2 June 2012 (UTC)
replies: text: Then perhaps these unsolved problems should just be incorporated into the article text and the template be removed entirely. Just because it has passed 4 TfDs is not an endorsement of the templates presence in this article. -- Toshio Yamaguchi (tlk−ctb) 07:19, 3 June 2012 (UTC)
replies: text: I think that would be the best solution. Ryan Vesey Review me! 07:27, 3 June 2012 (UTC)
thread_title:  Replacing Unsolved template 

message: Somewhere in the article should be the factorization of 1092 and 3510. The question is where? - Virginia-American (talk) 10:41, 25 September 2012 (UTC)
replies: text: I would suggest rename section 'Binary periodicity of p-1' with 'Properties of p-1' and show factorizations there. Maxal (talk) 01:12, 26 September 2012 (UTC)
replies: text: I don't think the prime factorizations of those two numbers are particularly interesting, as I guess that's what is supposed to be added. More interesting (in my opinion) are the divisors of those two numbers (see the section An unexpected property of the divisor-sums here). -- Toshio Yamaguchi (tlk−ctb) 13:17, 27 September 2012 (UTC)
thread_title:  p−1 

message: When I search the interval [1000, 1100] with wwww and tell it to report special instances with |A| ≤ 1000 I get
1009 is a special instance (+1 +296 p)
1009 is a special instance (+1 -713 p)
1013 is a special instance (-1 +41 p)
1013 is a special instance (-1 -972 p)
1019 is a special instance (-1 +657 p)
1019 is a special instance (-1 -362 p)
1021 is a special instance (-1 +644 p)
1021 is a special instance (-1 -377 p)
1031 is a special instance (+1 +318 p)
1031 is a special instance (+1 -713 p)
1033 is a special instance (+1 +251 p)
1033 is a special instance (+1 -782 p)
1039 is a special instance (+1 +872 p)
1039 is a special instance (+1 -167 p)
1049 is a special instance (+1 +798 p)
1049 is a special instance (+1 -251 p)
1051 is a special instance (-1 +845 p)
1051 is a special instance (-1 -206 p)
1061 is a special instance (-1 +880 p)
1061 is a special instance (-1 -181 p)
1063 is a special instance (+1 +297 p)
1063 is a special instance (+1 -766 p)
1069 is a special instance (-1 +978 p)
1069 is a special instance (-1 -91 p)
1087 is a special instance (+1 +975 p)
1087 is a special instance (+1 -112 p)
1091 is a special instance (-1 +386 p)
1091 is a special instance (-1 -705 p)
1093 is a Wieferich prime
1097 is a special instance (+1 +825 p)
1097 is a special instance (+1 -272 p)
However when I repeat the same with wwwwcl I get
1009 is a special instance (+1 -713 p)
1013 is a special instance (-1 -972 p)
1019 is a special instance (-1 -362 p)
1021 is a special instance (-1 -377 p)
1031 is a special instance (+1 -713 p)
1033 is a special instance (+1 -782 p)
1039 is a special instance (+1 -167 p)
1049 is a special instance (+1 -251 p)
1051 is a special instance (-1 -206 p)
1061 is a special instance (-1 -181 p)
1063 is a special instance (+1 -766 p)
1069 is a special instance (-1 -91 p)
1087 is a special instance (+1 -112 p)
1091 is a special instance (-1 -705 p)
1093 is a Wieferich prime
1097 is a special instance (+1 -272 p)
Now I don't know whether there's something wrong with wwwwcl, but I guess there are indeed two A values for each p and wwwwcl simply omits the larger one. -- Toshio Yamaguchi (tlk−ctb) 19:43, 28 October 2012 (UTC)
thread_title:  Mhm.... 

message: The last sentence in the section Wieferich prime#Periods of Wieferich primes currently reads
"Garza and Young claim that the period of 1093 were 1092 and that this were the same as the period of 10932,[52]:314 although the fact that the multiplicative order of 2 modulo 10932 is 364 shows that this is not the case."
I am unsure about whether that sentence should be kept or not. I am the editor who originally added that sentence. However, I don't know whether it is appropriate. While I believe the Garza & Young paper is what Wikipedia generally considers a reliable source, I don't know whether that is a reason to repeat that incorrect claim in the article. It seems to be just one particular error in an otherwise reliable source. The note I added to the end of the sentence might come close to WP:OR territory, although I don't know whether a claim such as this one needs a source (I don't know how likely it is  that the statement that the multiplicative order of 2 modulo 10932 is 364 will be challenged). -- Toshio Yamaguchi 11:54, 27 March 2013 (UTC)
replies: text: I would remove whole section on Periods of Wieferich primes as it repeats what is already said, just in different (recreational math) terms. I am also not sure if Mathematics Magazine is a reliable academic source. For me it sounds like a journal on recreational mathematics, with no so strict reviewing policy. So errors in obvious mathematical facts do not come as a surprise there. Maxal (talk) 01:23, 28 March 2013 (UTC)
replies: text: I removed the whole section. I agree that Mathematics Magazine doesn't appear to be a reliable academic source. http://www.maa.org/pubs/mm-guide.html says it is an expository journal of undergraduate mathematics and not a research journal. -- Toshio Yamaguchi 06:27, 28 March 2013 (UTC)
thread_title:  Statement about period of 1093 

message: Maybe this article needs a real shake-up. Also in 3.3 is the claim
::For the primes 1093 and 3511 it was shown that neither of them is a factor of any Mersenne or Fermat number.
This is befuddling because, for every prime p there is a divisor d of p-1 so that p divides exactly the Mersenne numbers  M_n with n a multiple of d.
In particular the numbers are d=364  for 1093 and d=1755  for 3511
I think that the article could be changed to read
::For the primes 1093 and 3511 it was shown that neither of them is a factor of any Mersenne number whose index is prime or any Fermat number (at all).
That would make sense with things said a little earlier.
The paper used as a reference is from a good journal and appears to say exactly what is claimed (without my addition). 
:: THEOREM 2. Neither 1093 nor 3511 divides any F_n or any M_q.
However the paper says at the start 
::Throughout, "p" and "q" will denote odd primes.
Still, this problem (and the other) makes me suspicious about the article as a whole.
What a Mersenne number is (perhaps only 2^n-1 for n odd? which would make the claim true..) is never explicitly stated. However it follows implicitly from the description of 2^n-1 as the nth Mersenne number that n can be even or odd.
--Gentlemath (talk) 06:33, 23 November 2013 (UTC)
replies: text: The paper by Bray and Warren explicitly talks about Mersenne numbers with prime index, so I adjusted the article to explicitly state that (diff). -- Toshio Yamaguchi 13:28, 23 November 2013 (UTC)
thread_title:  something else is wrong 

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thread_title:  External links modified 

message: In the section on the connections with the abc conjecture, there is the following sentence
"The set of Wieferich primes and the set of non-Wieferich primes, sometimes denoted by W2 and W2c respectively, are complementary sets, so if one of them is shown to be finite, the other one would necessarily have to be infinite, because both are proper subsets of the set of prime numbers."
The final clause seems to not be relevant to the point being made - the fact that they are subsets is (implicitly) implied by complementary, and whether they are proper subsets or not makes no difference as to whether one needs to be finite or not.
Instead the reason is that they are complementary and there are infinitely many primes. I propose replacing the final clause with one to this effect, or if this is deemed not to be at the appropriate level given the surrounding material, simply omitting the clause. 5.151.13.164 (talk) 18:43, 10 April 2024 (UTC)
replies: text: I removed the "because both" clause. —David Eppstein (talk) 20:36, 10 April 2024 (UTC)
thread_title:  Semi-protected edit request on 10 April 2024