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The following theorem connecting Wieferich primes and Fermat's last theorem was proven by Wieferich in 1909:
In
1910,
Mirimanoff was able to expand the theorem by showing that, if the preconditions of the theorem hold true for some prime p, then p² must divide mp − 1-1.
He proved for m=3.
Taro Morishima proved in
1931 for every prime number m not exceeding 31.
—Preceding unsigned comment added by 218.133.184.93 ( talk) 14:54, 26 January 2007
Wieferich primes and Fermat numbers.
This line has been added frequently, with the reference added to the "further reading" section. This is
— Arthur Rubin (talk) 19:28, 26 April 2008 (UTC)
Surely those were References not Further Reading? Is the Silverman reference used in the text? Richard Pinch ( talk) 07:05, 2 August 2008 (UTC)
I noticed that reference 60 ( http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html) gives an error 403. HannsEwald ( talk) 00:25, 14 October 2014 (UTC)
Is the W. Meissner who found the Wieferich prime 1093 really the physicist Walter Meissner? Richard Pinch ( talk) 07:06, 2 August 2008 (UTC)
Who conjectured that there were only finitely many Wieferich primes?
It seems to me that if you consider 2 to be a random element of , then 2p − 1 will be 1 mod p2 with probability 1/p, so that the expectation of the number of Wieferich primes less than x would be which is approximately log(log x), and this grows very slowly but grows to infinity nonetheless. Experiments along the lines of replacing 2 by various other small numbers supports this hueristic.
Johnny Vogler ( talk) 03:16, 15 October 2008 (UTC)
Considering 2 as a random element does not make sense; what is considered random here is 2p-1 modulo p2. More precisely, a heuristic argument here is the following assumption:
This assumption together with implies that
Maxal ( talk) 01:27, 9 June 2009 (UTC)
First, the group must be - everything is done modulo p2. Second, it sounds weird - how would you describe distribution of a ″random″ element 2 ? Third, assumption of the uniform distribution (in whatever form) is important here, otherwise there is no way to derive the probability of desired event. Maxal ( talk) 01:39, 9 June 2009 (UTC)
btw, ″any good number-theorist″ is not an argument. Maxal ( talk) 01:47, 9 June 2009 (UTC)
″it seems the proper formulation is that 2 is a random element″ - prove it, what is a distribution here? Moreover, "proper" or not is an issue of sense. ″I worked with Erdos″ is neither an argument. If you want this kind of argumentation - I have a Ph.D. in computer science - so what? Maxal ( talk) 01:55, 9 June 2009 (UTC)
Above I explained precisely what is motivation of my changes, while you did not provide any arguments besides ″any good number-theorist can see″ and ″I worked with Erdos″. Maxal ( talk) 01:59, 9 June 2009 (UTC)
OK. Let's cool down. The argument ″it does not make sense″ was exactly mine with respect to treating 2 as a random element. I still do not see what exactly that is supposed to mean. What I suggested may not be complete either but I will try to formalize it further later on. Maxal ( talk) 02:45, 9 June 2009 (UTC)
It seems to be simpler to use a somewhat reverse assumption - that the (p-1)-th degree roots of unity modulo p2 behave as uniformly distributed random elements in . Since the total number of these roots is p-1, the probability that 2 is one of them is
Maxal ( talk) 03:18, 9 June 2009 (UTC)
The argument that the Fermat quotient of p should be divisible by p with probability 1/p, leading to the heuristic estimate that the number of Wieferich primes below a given limit n should be asymptotic to log log n, appears in Andrew Granville, "Some Conjectures related to Fermat's Last Theorem," in Richard A. Mollin, ed., Number Theory (1990), pp. 177-92, at p. 178; available online at http://www.dms.umontreal.ca/~andrew/PDF/ConjFLT.pdf. John Blythe Dobson ( talk) 21:01, 9 March 2011 (UTC)
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 [4]). 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)
Ok, sure you're right with this, but just let me explain what my intention is. I would like to add a section about the Near-Wieferich primes to the article and a list of all known examples. As Near-Wieferich primes are commonly defined by the congruence 2(p-1)/2 ≡ ±1 + Ap (mod p2), it would be easier to understand the connection between Wieferich and Near-Wieferich primes. Toshio Yamaguchi ( talk) 15:16, 5 July 2010 (UTC)
I added the section about Near-Wieferich primes. Toshio Yamaguchi ( talk) 19:54, 5 July 2010 (UTC)
The citations given are not fully consistent with the citation style used in the rest of the article, as I had some problems generating those reflist citations. Maybe I'm a bit stupid, but can someone explain how I get a citation with these small numbers. As soon as I got this I will change the citations for consistency's sake. Toshio Yamaguchi ( talk) 21:04, 5 July 2010 (UTC)
I removed the reference to the Dorais&Klyve paper in the Near-Wieferich prime section, as they seem to use a modified version of the Near-Wieferich congruence in their paper and thus seem to get other Near-Wieferich prime values than the other searches. Instead I used the paper by Crandall, Dilcher and Pomerance as well as the paper by Knauer and Richstein as reference, as they seem to use the unaltered definition, which is also used in the current search by Wieferich@Home. Toshio Yamaguchi ( talk) 09:41, 6 July 2010 (UTC)
The section about the connection between Wieferich primes and Fermat's last theorem currently says:
"The following theorem connecting Wieferich primes and Fermat's last theorem was proven by Wieferich in 1909:
In 1910, Mirimanoff was able to expand the theorem by showing that, if the preconditions of the theorem hold true for some prime p, then p2 must also divide 3p − 1 − 1."
This reads as if when a prime p satisfies
then it must also satisfy
.
This however is not true as 1093 and 3511 only satisfy
and 11 and 1006003 only satisfy
but none of these primes satisfies both congruences. Am I missing something here? Toshio Yamaguchi ( talk) 18:40, 15 March 2011 (UTC)
I added a table containing all known near-Wieferich primes to the article a while ago. The reason for this is, that most of the recent searches for Wieferich primes also spent a considerable amount of computing power for searching near-Wieferich primes. Furthermore, the Wieferich primes can be considered special cases of near-Wieferich primes. However, some of the results regarding near-Wieferich primes are unpublished. The latest published result is that by Knauer and Richstein. The results of all further search efforts are unpublished as of 2011. More specifically the results of the search to 3x1015 by P. Carlisle, R. Crandall and M. Rodenkirch are unpublished (although the search is mentioned in the given reference). Furthermore, the results of the search to 6.7x1015 by Dorais and Klyve are also unpublished. The near-Wieferich primes found by Dorais and Klyve are from the unpublished draft that can be downloaded from their website. The near-Wieferich primes in [1.25x1015, 3x1015] were also added by me, after I received the values from Mark Rodenkirch via email. I would like to hear the opinions of other editors about what to do with the near-Wieferich primes in this article. Toshio Yamaguchi ( talk) 12:06, 27 March 2011 (UTC)
I reduced the lenght of both tables of near-Wieferich primes and made both uncollapsible. Toshio Yamaguchi ( talk) 22:04, 28 March 2011 (UTC)
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)
For a cyclotomic generalization of the Wieferich property:(np − 1)/(n − 1) divisible by q2, there are solutions like
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)
Nathanson, M. B. (2000). Elementary methods in number theory. Springer. p. 187. ISBN 0-387-98912-9.
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)
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)
A non-mathematician writes... The lead says that they were "first described" in 1909, at which time the theorems were "already well known." I'm sure this makes sense to mathematicians, but it reads a bit oddly to the non-specialist. Tigerboy1966 ( talk) 11:59, 7 December 2011 (UTC)
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)
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 p2 divides Mp-1. Thus I believe the statement should perhaps be removed. Opinions? Toshio Yamaguchi ( talk) 15:41, 8 January 2012 (UTC)
GA toolbox |
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Reviewer: GreatOrangePumpkin ( talk · contribs) 10:46, 7 May 2012 (UTC)
(Note): My common practice is to paste the template into the GAN. I often check points 5 and 6 (stability and pictures) before reviewing because they are easy to check. "Neutral" is confusing, as it may also mean that I did not review this points. Ok, now I will resume my review :)-- GoP T C N 11:14, 8 May 2012 (UTC)
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)
Wieferich_prime#Equivalent_congruences states that
But if k=1, you have
which is true by definition. Squaring it yields
which is clearly not 2.
The best I've come up with yet is
and this means that k=p gives the result
which simplifies to
I tried to prove you wrong, but that's actually the result which CAN be found here. Can anyone please clarify the steps in that sentence, Thus a Wieferich prime satisfies (...) for all integers k ≥ 1., and tell me where I did wrong, I feel my neurons melt when thinking about it any longer. Never was my signature any truer, I'm afraid. - ¡Ouch! ( hurt me / more pain) 11:07, 7 September 2012 (UTC)
get 2p2 ≡ 2 (mod p2). -- Toshio Yamaguchi ( tlk− ctb) 11:57, 7 September 2012 (UTC)
(Sorry Toshio for butchering your reply, but it took me 5 minutes to spot the error, and IMO this is one of the rare cases where editing you makes the issue easier to understand for third parties. I didn't change your words, but pointed out which equations are off. Not meant as humiliation, but to save other readers 5 minutes of their time.)
I found an approach which proves above equation without damaging my synapses. It goes as follows:
I take the following equations,
Once I realized that pn - 1 is a multiple of (p-1), all
because they are powers of 2p-1. Now multiply by Eq2 to get the result,
Maybe this could go into the article, as it is easier to grasp IMO than what is in there now. Comments? - ¡Ouch! ( hurt me / more pain) 08:14, 10 September 2012 (UTC)
It's not true in general. bp^2-p = 1, (mod p2) is the general case. For example 2^25 = 33554432, is equal to 7, mod 25, not 2. Likewise, 3^9 = 512 which is 8, mod 9, not the value 2. By fermat's little theorm, b^(p-1) = 1, modulo p, so we get b^n(p-1)+1 = b, modulo p. For all primes, b^(p^2-p) = 1, mod p² Wendy.krieger ( talk) 11:15, 5 October 2018 (UTC)
Oops. I think the "Equivalent congruences" section is wrong again..
I get 2p2 = (2p)p ≡ 2p (mod p2) instead.
Which reduces to 2. Induction yields 2pk ≡ 2 (mod p2) for all positive integers k. - ¡Ouch! ( hurt me / more pain) 09:29, 14 September 2012 (UTC)
The fact that pk - pk - 1 is a multiple of (p - 1) proves that p, p2, ... , pk all reduce to 1 (modulo (p - 1)).
Which in turn proves that all 2pkare
2 times a power of (2p - 1 ) (mod p2),
the latter factor being 1 by definition.
I hope. {Headache} - ¡Ouch! ( hurt me / more pain) 09:29, 14 September 2012 (UTC)
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)
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)
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)
I know that for any prime number p, the expression in that section adds up to a number > 0 and < p2 and so is not a multiple of p2. I always thought ≡ a mod m meant it is m * an integer + a. Is that only the definition for when both a and m are integers? In that context, what does the article really mean by ≡ 0 mod p2? There were also so few terms listed in the expression that it's unclear whether it means the reciprocals of all odd numbers up to p-2. Blackbombchu ( talk) 02:28, 3 October 2013 (UTC)
That's right, for any prime number p, that series will eventually pass p but only after you get to the reciprocal of an odd number much higher than p that's even larger than 2p. Blackbombchu ( talk) 19:10, 3 October 2013 (UTC)
Section 3.3 says
One problem is that M6k is always a multiple of 9 so it can not be the case that there are at most finitely many Mersenne numbers that are not square-free. (aside: wouldn't "only" make better sense than "at most"?)
I was about to change it to
because that seems to go with what came before and be true.
Then I noticed a second problem: the converse of the current claim would be
equivalently:
But that is not what it says.
So I decided to hold back in hopes that someone could clarify this. With some difficulty I got through to the article but my French is a little rusty. I looked in the review in math reviews which says (after hacking the TeX slightly)
-- Gentlemath ( talk) 05:58, 23 November 2013 (UTC)
Maybe this article needs a real shake-up. Also in 3.3 is the claim
This is befuddling because, for every prime there is a divisor of so that divides exactly the Mersenne numbers with a multiple of .
In particular the numbers are for and for
I think that the article could be changed to read
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).
However the paper says at the start
Still, this problem (and the other) makes me suspicious about the article as a whole.
What a Mersenne number is (perhaps only for odd? which would make the claim true..) is never explicitly stated. However it follows implicitly from the description of as the th Mersenne number that can be even or odd.
-- Gentlemath ( talk) 06:33, 23 November 2013 (UTC)
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Please check for and forgive any errors although I've double-checked myself. I do not consider myself a mathematician. The following is merely a suggestion that makes a lot of sense to me but which might be entirely inappropriate for Wikipedia.
The "Equivalent definitions" section shows some detail on the the Fermat quotient with p = 1093 but nowhere in the entire article is the actual Wieferich prime calculation for p = 1093 shown where one divides by p^2 = 1194649.
I assume this was done by hand over a hundred years ago so why not simply do the following somewhere early in the article to show everybody at least one example (out of the two known Wieferich primes) in the most direct and unambiguous way possible of what the article is actually about:
p = 1093
p-1 = 1092
p^2 = 1194649
2^(p-1)-1 =
5305853629099163473739654017028385853919897839577127147455154959874 2698935712572215646062825029533563666321339663466341699370021653261 4451998263607140649559448662636695562122335268130631432841045579576 5830555928325311888973488264278654408606329328273740531145437577728 5526890640894984855797452638426498665888535738081213096656895
(2^(p-1)-1) / (p^2) =
4441349408151819884953366233118167640804870585064840926042004772844 8020243362336732919931147165011282532627859449483774480512704278211 7971051131844701372168267551922527505670983919235383307432597842191 7927823091406188670457589019267294752355151453082654847696216694383 4989934818423641467742786909315203600294760836095968855
Why not include this? It is not particularly long compared to the length of the article and please do not truncate the relatively small long numbers when they only span a handful of lines.
Is the equivalence more important than what it is equivalent to? 90.149.36.98 ( talk) 14:41, 10 August 2018 (UTC)
From the article:
“A Wieferich prime base a is a prime p that satisfies
Such a prime cannot divide a, since then it would also divide 1.
It's a conjecture that for every natural number a, there are infinitely many Wieferich primes in base a.”
Is there a reason for the "subsequence of" part to be in a serif font? Woah! // Talk? 13:15, 22 December 2022 (UTC)
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The following is a suggested change to the /info/en/?search=Wieferich_prime#Wieferich_sequence section.
The following line:
2, 1093, 5, 20771, 18043, 5, 20771, 18043, 5, ..., it gets a cycle: {5, 20771, 18043}.
should be changed to:
2, 1093, 5, 20771, 18043, 5, 20771, 18043, 5, ... (sequence A359952 in the OEIS), it gets a cycle: {5, 20771, 18043}.
Thank you. Boblyonsnj ( talk) 17:12, 28 January 2023 (UTC)
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)
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The following theorem connecting Wieferich primes and Fermat's last theorem was proven by Wieferich in 1909:
In
1910,
Mirimanoff was able to expand the theorem by showing that, if the preconditions of the theorem hold true for some prime p, then p² must divide mp − 1-1.
He proved for m=3.
Taro Morishima proved in
1931 for every prime number m not exceeding 31.
—Preceding unsigned comment added by 218.133.184.93 ( talk) 14:54, 26 January 2007
Wieferich primes and Fermat numbers.
This line has been added frequently, with the reference added to the "further reading" section. This is
— Arthur Rubin (talk) 19:28, 26 April 2008 (UTC)
Surely those were References not Further Reading? Is the Silverman reference used in the text? Richard Pinch ( talk) 07:05, 2 August 2008 (UTC)
I noticed that reference 60 ( http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html) gives an error 403. HannsEwald ( talk) 00:25, 14 October 2014 (UTC)
Is the W. Meissner who found the Wieferich prime 1093 really the physicist Walter Meissner? Richard Pinch ( talk) 07:06, 2 August 2008 (UTC)
Who conjectured that there were only finitely many Wieferich primes?
It seems to me that if you consider 2 to be a random element of , then 2p − 1 will be 1 mod p2 with probability 1/p, so that the expectation of the number of Wieferich primes less than x would be which is approximately log(log x), and this grows very slowly but grows to infinity nonetheless. Experiments along the lines of replacing 2 by various other small numbers supports this hueristic.
Johnny Vogler ( talk) 03:16, 15 October 2008 (UTC)
Considering 2 as a random element does not make sense; what is considered random here is 2p-1 modulo p2. More precisely, a heuristic argument here is the following assumption:
This assumption together with implies that
Maxal ( talk) 01:27, 9 June 2009 (UTC)
First, the group must be - everything is done modulo p2. Second, it sounds weird - how would you describe distribution of a ″random″ element 2 ? Third, assumption of the uniform distribution (in whatever form) is important here, otherwise there is no way to derive the probability of desired event. Maxal ( talk) 01:39, 9 June 2009 (UTC)
btw, ″any good number-theorist″ is not an argument. Maxal ( talk) 01:47, 9 June 2009 (UTC)
″it seems the proper formulation is that 2 is a random element″ - prove it, what is a distribution here? Moreover, "proper" or not is an issue of sense. ″I worked with Erdos″ is neither an argument. If you want this kind of argumentation - I have a Ph.D. in computer science - so what? Maxal ( talk) 01:55, 9 June 2009 (UTC)
Above I explained precisely what is motivation of my changes, while you did not provide any arguments besides ″any good number-theorist can see″ and ″I worked with Erdos″. Maxal ( talk) 01:59, 9 June 2009 (UTC)
OK. Let's cool down. The argument ″it does not make sense″ was exactly mine with respect to treating 2 as a random element. I still do not see what exactly that is supposed to mean. What I suggested may not be complete either but I will try to formalize it further later on. Maxal ( talk) 02:45, 9 June 2009 (UTC)
It seems to be simpler to use a somewhat reverse assumption - that the (p-1)-th degree roots of unity modulo p2 behave as uniformly distributed random elements in . Since the total number of these roots is p-1, the probability that 2 is one of them is
Maxal ( talk) 03:18, 9 June 2009 (UTC)
The argument that the Fermat quotient of p should be divisible by p with probability 1/p, leading to the heuristic estimate that the number of Wieferich primes below a given limit n should be asymptotic to log log n, appears in Andrew Granville, "Some Conjectures related to Fermat's Last Theorem," in Richard A. Mollin, ed., Number Theory (1990), pp. 177-92, at p. 178; available online at http://www.dms.umontreal.ca/~andrew/PDF/ConjFLT.pdf. John Blythe Dobson ( talk) 21:01, 9 March 2011 (UTC)
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 [4]). 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)
Ok, sure you're right with this, but just let me explain what my intention is. I would like to add a section about the Near-Wieferich primes to the article and a list of all known examples. As Near-Wieferich primes are commonly defined by the congruence 2(p-1)/2 ≡ ±1 + Ap (mod p2), it would be easier to understand the connection between Wieferich and Near-Wieferich primes. Toshio Yamaguchi ( talk) 15:16, 5 July 2010 (UTC)
I added the section about Near-Wieferich primes. Toshio Yamaguchi ( talk) 19:54, 5 July 2010 (UTC)
The citations given are not fully consistent with the citation style used in the rest of the article, as I had some problems generating those reflist citations. Maybe I'm a bit stupid, but can someone explain how I get a citation with these small numbers. As soon as I got this I will change the citations for consistency's sake. Toshio Yamaguchi ( talk) 21:04, 5 July 2010 (UTC)
I removed the reference to the Dorais&Klyve paper in the Near-Wieferich prime section, as they seem to use a modified version of the Near-Wieferich congruence in their paper and thus seem to get other Near-Wieferich prime values than the other searches. Instead I used the paper by Crandall, Dilcher and Pomerance as well as the paper by Knauer and Richstein as reference, as they seem to use the unaltered definition, which is also used in the current search by Wieferich@Home. Toshio Yamaguchi ( talk) 09:41, 6 July 2010 (UTC)
The section about the connection between Wieferich primes and Fermat's last theorem currently says:
"The following theorem connecting Wieferich primes and Fermat's last theorem was proven by Wieferich in 1909:
In 1910, Mirimanoff was able to expand the theorem by showing that, if the preconditions of the theorem hold true for some prime p, then p2 must also divide 3p − 1 − 1."
This reads as if when a prime p satisfies
then it must also satisfy
.
This however is not true as 1093 and 3511 only satisfy
and 11 and 1006003 only satisfy
but none of these primes satisfies both congruences. Am I missing something here? Toshio Yamaguchi ( talk) 18:40, 15 March 2011 (UTC)
I added a table containing all known near-Wieferich primes to the article a while ago. The reason for this is, that most of the recent searches for Wieferich primes also spent a considerable amount of computing power for searching near-Wieferich primes. Furthermore, the Wieferich primes can be considered special cases of near-Wieferich primes. However, some of the results regarding near-Wieferich primes are unpublished. The latest published result is that by Knauer and Richstein. The results of all further search efforts are unpublished as of 2011. More specifically the results of the search to 3x1015 by P. Carlisle, R. Crandall and M. Rodenkirch are unpublished (although the search is mentioned in the given reference). Furthermore, the results of the search to 6.7x1015 by Dorais and Klyve are also unpublished. The near-Wieferich primes found by Dorais and Klyve are from the unpublished draft that can be downloaded from their website. The near-Wieferich primes in [1.25x1015, 3x1015] were also added by me, after I received the values from Mark Rodenkirch via email. I would like to hear the opinions of other editors about what to do with the near-Wieferich primes in this article. Toshio Yamaguchi ( talk) 12:06, 27 March 2011 (UTC)
I reduced the lenght of both tables of near-Wieferich primes and made both uncollapsible. Toshio Yamaguchi ( talk) 22:04, 28 March 2011 (UTC)
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)
For a cyclotomic generalization of the Wieferich property:(np − 1)/(n − 1) divisible by q2, there are solutions like
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)
Nathanson, M. B. (2000). Elementary methods in number theory. Springer. p. 187. ISBN 0-387-98912-9.
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)
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)
A non-mathematician writes... The lead says that they were "first described" in 1909, at which time the theorems were "already well known." I'm sure this makes sense to mathematicians, but it reads a bit oddly to the non-specialist. Tigerboy1966 ( talk) 11:59, 7 December 2011 (UTC)
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)
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 p2 divides Mp-1. Thus I believe the statement should perhaps be removed. Opinions? Toshio Yamaguchi ( talk) 15:41, 8 January 2012 (UTC)
GA toolbox |
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Reviewing |
Reviewer: GreatOrangePumpkin ( talk · contribs) 10:46, 7 May 2012 (UTC)
(Note): My common practice is to paste the template into the GAN. I often check points 5 and 6 (stability and pictures) before reviewing because they are easy to check. "Neutral" is confusing, as it may also mean that I did not review this points. Ok, now I will resume my review :)-- GoP T C N 11:14, 8 May 2012 (UTC)
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)
Wieferich_prime#Equivalent_congruences states that
But if k=1, you have
which is true by definition. Squaring it yields
which is clearly not 2.
The best I've come up with yet is
and this means that k=p gives the result
which simplifies to
I tried to prove you wrong, but that's actually the result which CAN be found here. Can anyone please clarify the steps in that sentence, Thus a Wieferich prime satisfies (...) for all integers k ≥ 1., and tell me where I did wrong, I feel my neurons melt when thinking about it any longer. Never was my signature any truer, I'm afraid. - ¡Ouch! ( hurt me / more pain) 11:07, 7 September 2012 (UTC)
get 2p2 ≡ 2 (mod p2). -- Toshio Yamaguchi ( tlk− ctb) 11:57, 7 September 2012 (UTC)
(Sorry Toshio for butchering your reply, but it took me 5 minutes to spot the error, and IMO this is one of the rare cases where editing you makes the issue easier to understand for third parties. I didn't change your words, but pointed out which equations are off. Not meant as humiliation, but to save other readers 5 minutes of their time.)
I found an approach which proves above equation without damaging my synapses. It goes as follows:
I take the following equations,
Once I realized that pn - 1 is a multiple of (p-1), all
because they are powers of 2p-1. Now multiply by Eq2 to get the result,
Maybe this could go into the article, as it is easier to grasp IMO than what is in there now. Comments? - ¡Ouch! ( hurt me / more pain) 08:14, 10 September 2012 (UTC)
It's not true in general. bp^2-p = 1, (mod p2) is the general case. For example 2^25 = 33554432, is equal to 7, mod 25, not 2. Likewise, 3^9 = 512 which is 8, mod 9, not the value 2. By fermat's little theorm, b^(p-1) = 1, modulo p, so we get b^n(p-1)+1 = b, modulo p. For all primes, b^(p^2-p) = 1, mod p² Wendy.krieger ( talk) 11:15, 5 October 2018 (UTC)
Oops. I think the "Equivalent congruences" section is wrong again..
I get 2p2 = (2p)p ≡ 2p (mod p2) instead.
Which reduces to 2. Induction yields 2pk ≡ 2 (mod p2) for all positive integers k. - ¡Ouch! ( hurt me / more pain) 09:29, 14 September 2012 (UTC)
The fact that pk - pk - 1 is a multiple of (p - 1) proves that p, p2, ... , pk all reduce to 1 (modulo (p - 1)).
Which in turn proves that all 2pkare
2 times a power of (2p - 1 ) (mod p2),
the latter factor being 1 by definition.
I hope. {Headache} - ¡Ouch! ( hurt me / more pain) 09:29, 14 September 2012 (UTC)
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)
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)
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)
I know that for any prime number p, the expression in that section adds up to a number > 0 and < p2 and so is not a multiple of p2. I always thought ≡ a mod m meant it is m * an integer + a. Is that only the definition for when both a and m are integers? In that context, what does the article really mean by ≡ 0 mod p2? There were also so few terms listed in the expression that it's unclear whether it means the reciprocals of all odd numbers up to p-2. Blackbombchu ( talk) 02:28, 3 October 2013 (UTC)
That's right, for any prime number p, that series will eventually pass p but only after you get to the reciprocal of an odd number much higher than p that's even larger than 2p. Blackbombchu ( talk) 19:10, 3 October 2013 (UTC)
Section 3.3 says
One problem is that M6k is always a multiple of 9 so it can not be the case that there are at most finitely many Mersenne numbers that are not square-free. (aside: wouldn't "only" make better sense than "at most"?)
I was about to change it to
because that seems to go with what came before and be true.
Then I noticed a second problem: the converse of the current claim would be
equivalently:
But that is not what it says.
So I decided to hold back in hopes that someone could clarify this. With some difficulty I got through to the article but my French is a little rusty. I looked in the review in math reviews which says (after hacking the TeX slightly)
-- Gentlemath ( talk) 05:58, 23 November 2013 (UTC)
Maybe this article needs a real shake-up. Also in 3.3 is the claim
This is befuddling because, for every prime there is a divisor of so that divides exactly the Mersenne numbers with a multiple of .
In particular the numbers are for and for
I think that the article could be changed to read
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).
However the paper says at the start
Still, this problem (and the other) makes me suspicious about the article as a whole.
What a Mersenne number is (perhaps only for odd? which would make the claim true..) is never explicitly stated. However it follows implicitly from the description of as the th Mersenne number that can be even or odd.
-- Gentlemath ( talk) 06:33, 23 November 2013 (UTC)
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Please check for and forgive any errors although I've double-checked myself. I do not consider myself a mathematician. The following is merely a suggestion that makes a lot of sense to me but which might be entirely inappropriate for Wikipedia.
The "Equivalent definitions" section shows some detail on the the Fermat quotient with p = 1093 but nowhere in the entire article is the actual Wieferich prime calculation for p = 1093 shown where one divides by p^2 = 1194649.
I assume this was done by hand over a hundred years ago so why not simply do the following somewhere early in the article to show everybody at least one example (out of the two known Wieferich primes) in the most direct and unambiguous way possible of what the article is actually about:
p = 1093
p-1 = 1092
p^2 = 1194649
2^(p-1)-1 =
5305853629099163473739654017028385853919897839577127147455154959874 2698935712572215646062825029533563666321339663466341699370021653261 4451998263607140649559448662636695562122335268130631432841045579576 5830555928325311888973488264278654408606329328273740531145437577728 5526890640894984855797452638426498665888535738081213096656895
(2^(p-1)-1) / (p^2) =
4441349408151819884953366233118167640804870585064840926042004772844 8020243362336732919931147165011282532627859449483774480512704278211 7971051131844701372168267551922527505670983919235383307432597842191 7927823091406188670457589019267294752355151453082654847696216694383 4989934818423641467742786909315203600294760836095968855
Why not include this? It is not particularly long compared to the length of the article and please do not truncate the relatively small long numbers when they only span a handful of lines.
Is the equivalence more important than what it is equivalent to? 90.149.36.98 ( talk) 14:41, 10 August 2018 (UTC)
From the article:
“A Wieferich prime base a is a prime p that satisfies
Such a prime cannot divide a, since then it would also divide 1.
It's a conjecture that for every natural number a, there are infinitely many Wieferich primes in base a.”
Is there a reason for the "subsequence of" part to be in a serif font? Woah! // Talk? 13:15, 22 December 2022 (UTC)
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The following is a suggested change to the /info/en/?search=Wieferich_prime#Wieferich_sequence section.
The following line:
2, 1093, 5, 20771, 18043, 5, 20771, 18043, 5, ..., it gets a cycle: {5, 20771, 18043}.
should be changed to:
2, 1093, 5, 20771, 18043, 5, 20771, 18043, 5, ... (sequence A359952 in the OEIS), it gets a cycle: {5, 20771, 18043}.
Thank you. Boblyonsnj ( talk) 17:12, 28 January 2023 (UTC)
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)