Comments posted on May 22, 2018

I thank Mercedes Orús-Lacort a lot for her comments and for the two references that she has posted on May 20, 2018 as answers to my question about the history of the first proof of Fermat Last Theorem (FLT).

I say the “first” proof, because I believe that it is worth the effort to try to find other proofs. Many mathematicians seem opposed to such attempts. I am not one of them.

I will add Mercedes’ contribution to the file about the history of the construction of “Victory Road” as soon as possible. (I am overflood with too many marvels in mathematics.)

Concerning Richard Taylor, there is no doubt that he has been a top student in mathematics, and after this a top mathematician, as you can see on the reference given by Mercedes. There are not so many graduate students who can be selected by Andrew Wiles as PhD students. After obtaining his PhD, Richard became a top expert in number theory. When a gap was discovered in the last step of the proof of FLT, it turned out that the gap was extremely difficult to solve. It seemed that it would take a very long time to solve it. It is obvious that no mathematician who has almost finished to solve one of the most difficult problems of mathematics is enthusiastic to ask for help. But solving the gap seemed so hopeless that Andrew wondered whether it may be better to collaborate with someone. Whom did Andrew select? The best of the bests: Richard. After this collaboration with Andrew, Richard was on the team to solve an another famous problem of mathematics of astronomic dimension, namely, the proof of the full Taniyama–Shimura conjecture. Andrew had already solved a special case sufficient for the proof of FLT. This special case was already at the superman level, but there was no hope to solve the full conjecture any time soon. Nevertheless, the team Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor did just that. In addition, Richard has received several top awards in mathematics.

There is an obvious question concerning the contribution of Richard to the final step of the construction of Victory Road. At the end, there was a gap. The gap seemed impossible to solve in a short time. After trying, and trying, and trying, and trying, Andrew decided to ask for the collaboration of Richard. They worked for several months. There is no doubt that with such top mathematicians, their work was very productive, related or not related to the gap in the proof of FLT. Then there is this posted on Wikipedia:

https://en.m.wikipedia.org/wiki/Fermat%27s_Last_Theorem

“Wiles states that on the morning of September 19, 1994, he was on the verge of giving up and was almost resigned to accepting that he had failed, and to publishing his work so that others could build on it and find the error. He adds that he was having a final look to try and understand the fundamental reasons why his approach could not be made to work, when he had a sudden insight that the specific reason why the Kolyvagin–Flach approach would not work directly also meant that his original attempts using Iwasawa theory could be made to work, if he strengthened it using his experience gained from the Kolyvagin–Flach approach. Fixing one approach with tools from the other approach would resolve the issue for all the cases that were not already proven by his refereed paper. He described later that Iwasawa theory and the Kolyvagin–Flach approach were each inadequate on their own, but together they could be made powerful enough to overcome this final hurdle.”

The obvious question is whether Andrew could have finished his proof on his own, without Richard, since he found how to solve the terrible gap without help.

Additional pieces of information:

http://www.math.ias.edu/~rtaylor/

https://en.m.wikipedia.org/wiki/Christophe_Breuil

https://en.m.wikipedia.org/wiki/Brian_Conrad

https://en.m.wikipedia.org/wiki/Fred_Diamond

https://en.m.wikipedia.org/wiki/Modularity_theorem

I thank Mercedes again a lot for her important contribution to the history of the proof. With best regards, Jean-Claude

Dear Jean-Claude,

You ask "whether Wiles could have finished his proof on his own, without Taylor, since he found how to solve the terrible gap without help." The strictly mathematical answer should be "yes" : what was needed to solve FLT was only (!) a particular case of the Shimura-Taniyama-Weil conjecture, the so called "semi-stable" case. As stated by Wiles himself, he came back to Iwasawa theory (which comprises the Kolyvagin-Flach method) as a final attempt, and it worked. But the act of creation depends on so many unpredictable factors that nobody can decide whether Wiles' collaboration with Taylor was or not a decisive step in the "rumination" process which led to Victory. What is certain is that Wiles' work (before and with Taylor) opened the path to the complete resolution by Breuil-Conrad-Diamond of the STW conjecture, now the Modularity Theorem for elliptic curves over Q, and later on to the proof by Khare-Winterberger of the Serre conjecture on modular representations (of which FLT is now a corollary which can be shown in half a page !)

Discussion on the history of Victory Road to the proof of FLT

Comments posted May 26, 2018

I am very grateful to Thong Nguyen Quang Do for her comments on the history of the proof of FLT, and for all the information. It seems very likely that, after all, Andrew did not need the help of Richard to construct the last part of Victory Road, namely, the semi-stable case of the Taniyama–Shimura–Weil Conjecture (TSWC).

I have looked for more information from the encyclopedia Wikipedia about the proof of the full TSWC:

https://en.wikipedia.org/wiki/Modularity_theorem

https://en.wikipedia.org/wiki/Modularity_theorem#History

The Modularity Theorem has been proved in the following 5 steps:

Step 1, May 1995:

Modular Elliptic Curves and Fermat's Last Theorem

Andrew Wiles

http://www.jstor.org/stable/2118559?seq=1#page_scan_tab_contents

Step 2, May 1995:

Ring-Theoretic Properties of Certain Hecke Algebras

Richard Taylor and Andrew Wiles

http://www.jstor.org/stable/2118560?seq=1#page_scan_tab_contents

Step 3, July 1996:

On Deformation Rings and Hecke Rings

Fred Diamond

http://www.jstor.org/stable/2118586?seq=1#page_scan_tab_contents

Step 4, 1999:

Modularity of certain potentially Barsotti-Tate Galois representations

Brian Conrad, Fred Diamond, and Richard Taylor

http://www.ams.org/journals/jams/1999-12-02/S0894-0347-99-00287-8/home.html

Step 5, May 2001:

On the modularity of elliptic curves over Q: wild 3-adic exercises",

Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor

http://www.ams.org/journals/jams/2001-14-04/S0894-0347-01-00370-8/home.html

This last step completed the proof of the full TSWC, which was no longer a conjecture, it was a theorem, namely, the Modularity Theorem for elliptic curves over Q.

I have also looked for more information from the encyclopedia Wikipedia about the proof of the Serre's modularity conjecture:

https://en.wikipedia.org/wiki/Serre%27s_modularity_conjecture

This conjecture was solved in the following 9 parts:

Part 1, December 5, 2004:

The level 1 weight 2 case of Serre's conjecture

Luis Victor Dieulefait

https://arxiv.org/abs/math/0412099v1

Part 2, December 19, 2004:

The level 1 weight 2 case of Serre's conjecture

Luis Victor Dieulefait

https://arxiv.org/abs/math/0412099v2

Part 3, February 22, 2005:

The level 1 weight 2 case of Serre's conjecture

Luis Victor Dieulefait

https://arxiv.org/abs/math/0412099v3

Part 4, 2006:

Serre's modularity conjecture: The level one case

Chandrashekhar Khare

https://projecteuclid.org/euclid.dmj/1156771903

Part 5, February 22, 2005:

The level 1 weight 2 case of Serre's conjecture

Luis Victor Dieulefait

https://arxiv.org/abs/math/0412099v4

https://arxiv.org/abs/math/0412099

Part 6, 2007:

The level 1 weight 2 case of Serre's conjecture

Luis Victor Dieulefait

http://www.ems-ph.org/journals/show_abstract.php?issn=0213-2230&vol=23&iss=3&rank=15

Part 7, July 4, 2009:

Serre's modularity conjecture, part 1

Chandrashekhar Khare and Jean-Pierre Wintenberger

https://link.springer.com/article/10.1007%2Fs00222-009-0205-7

Part 8, July 4, 2009:

Serre's modularity conjecture, part 2

Chandrashekhar Khare and Jean-Pierre Wintenberger

https://link.springer.com/article/10.1007%2Fs00222-009-0206-6

Part 9, 2009:

On Serre's reciprocity conjecture for 2-dimensional mod p representations of Gal(Q/Q)

Chandrashekhar Khare and Jean-Pierre Wintenberger

http://annals.math.princeton.edu/2009/169-1/p05

I have a lot to update in my files. There is so much work to do in so many directions that I need one million hours per day to do all this.

With a lot of thanks again to, I am not sure which one is the best, Thong, or Thong Nguyen, or Thong Nguyen Quang, or Thong Nguyen Quang Do.

With best regards, Jean-Claude

@ Jean-Claude Evrard

Thong is my given name, Nguyen Quang Do my family name... and I'm not a "her", what you see is the picture of my grand daughter !

Discussion on the history of Victory Road to the proof of FLT

Second comments posted on May 26, 2018

I thank Thong Nguyen Quang Do for his help about given name, family name, and picture.

While the proof of the extension from the semi-stable case to the full Shimura-Taniyama-Weil Conjecture (STWC) is not a part of Victory Road to the proof of FLT, Victory Road is a part of Super Victory Road to the super proof of the super conjecture STW.

I have just found the following reference to an article published in 1999 announcing the proof of the full Shimura-Taniyama-Weil Conjecture for all elliptic curves over Q, and summarizing the conjecture and its proof:

===========================================

A Proof of the Full Shimura-Taniyama-Weil Conjecture is Announced

Henri Darmon

Notices of the American Mathematical Society

Volume 46, issue 11, December 1999

http://www.ams.org/cgi-bin/notices/nxg-issue?year=1999&issue=12

Free pdf copy:

http://www.ams.org/notices/199911/comm-darmon.pdf

===========================================

With best regards, Jean-Claude

Dear Jean-Claude,

There has been quite a few expository articles on the subject, and also 2 or 3 books of proceedings.

I append here a certain number of accounts of Wiles' proof.

As well as references to some books of proceedings of conferences or seminars (high level)

[C] "Modular Forms and FLT", ed. G. Cornell, Spriger 1997

[Mu] "Seminar on FLT", ed. V. Kumar Murthy, AMS 1995

I add the "lower level" book by Hellegouarch because he was actually the inventor (thesis,1972) of the so called Frey curve, but at that time, the theory of modular forms was still embryonic; see also the Fermat-Hell. pdf.

[H] Y. Hellegouarch, "Introduction aux mathématiques de Fermat-Wiles", Masson 1997

At the popularization level, see

[Mo] C.J. Mozzochi, "The Fermat diary", AMS 2000

[S] S. Singh"FLT-The story of a riddle...", 4th Estate, 1997

The book [S] is rather anecdotical and contains inaccuracies (for instance S. makes a confusion between elliptic curves and elliptic functions), but [Mo] could be a serious contender for your project.

Monday, May 28, 2018

I am very grateful to Thong Nguyen Quang Do for all the links to wonderful articles related to the first proof of FLT. These articles are very informative to everyone working in other areas and who wants to have a broader view of what is going on in mathematics.

There is a lot to read. I have just started at random with the article published by Karl Rubin and Alice Silverberg on July 1994, before the proof of FLT was completed. This article explains the mathematics involved in the historic lecture given by Andrew Wiles on June 23, 1993 at the Newton Institute in Cambridge:

FLT Rubin.pdf

https://www.researchgate.net/profile/Thong_Nguyen_Quang_Do3/post/Are_there_other_pieces_of_information_about_Victory_Road_to_FLT/attachment/5b0abd964cde260d15e13fd9/AS%3A630903608991745%401527430550381/download/FLT+Rubin.pdf

https://arxiv.org/pdf/math/9407220.pdf

This article explains the mathematics involved in the historic lecture given by Andrew Wiles on June 23, 1993 at the Newton Institute in Cambridge.

I have had a first look at several small parts of this article.

Every time, I have been very impressed by the clarity and the high quality of writing.

For non-experts, I cannot recommend this article enough. It is immensely instructive.

With a lot of thanks to Thong for his huge contribution to this discussion, and with best regards, Jean-Claude

@ Mohamed Azzedine

The reviewers of "elementary proofs" of FLT have a trick, called the "Krasner trick", which allows them to decide at one glance whether the proof is blatantly incorrect. Yours is, unfortunately. For details on the subject (and more discussion - unfortunately !), see on RG

P.N.Seetharaman Seetharaman

How mathematicians view my published paper " A proof of Fermat's last theoem using an Euler's equation" ?

I don't pretend that your answer has something to do with S. S. As I said, the Krasner trick allows to see whether an elementary proof of FLT can succeed or not *without* even looking at details. Your "solution" here is the ideal example for a test. I don't intend to explain the trick here, just look at one of my (few) answers to S. S. and try to understand.

Please read my FLT Proof.

https://www.researchgate.net/publication/315721760_Fermat_Last_Theoreme_Demostration

My first reflex at Mr PSN's announcement (and his announcement alone) was "Why should I react to a confidential publication which I'm not aware of" ? But then it appeared progressively that the author posted his article in two public RG threads (*), out of vanity or in search of publicity. I felt compelled to take a look, because if PSN's paper is incorrect (and it is) and I keep silent, many "fermatist amateurs" would happily cry out at "double standards" ! So I took a look, just to see that my ancient fundamental objection was still there, alive and well, if I can say so: 3) By trial and error method, we have created the transformation equations...etc. I'm obliged to react publicly on RG (*), because this is again the same extra non necessary condition, amounting to a tautology, of which we have discussed at length, time and again, and I have no intention to come back to it. I don't even want to invoke again the "Krasner trick", nor "high brow" arguments. Just "elementary" ones, as the reader can check, and ask the adverse parties simply to refute them.

(*) See the 2 threads:  "Are there other pieces of information about Victory Road to FLT" and "Is it possible now to say that FLT has other proofs from other people besides the famous proof of Andrew Wiles"

* You did not read carefully my objection. I said that what you call "transformed equations" by error and trial in 3) is not a necessary condition. But we have discussed of this many times, so allow me not to begin again..

Come on, Mr PSN, even good willed readers get tired at your blind obstination. Let me specifically address the beginning of your « proof », where « by trial and error » you create a kind of Pythagorean equation which you use to « transform » the equations of FLT_3 and FLT_p . Not even insisting on the « transformations », a non sensical word if no precise definition is given (as pointed out by Prof. Farid), let’s examine logically your feat of prestidigitation :

The subject is FLT_n, but you « transform » it into a system of two equations FLT_n and FLT_3 . If this is not adding a non necessary condition, tell us what it is ? If you visualize curves better than surfaces, just replace the 3 variable equation FLT_n with integer unknowns x,y,z by the 2 variable equation X^n + Y^n = 1 with rational unknowns (same thing for n=3) ; what you do is to is to reduce the search for rational points of the curve X^n + Y^n = 1 to the search for rational intersection points of the two curves X^n + Y^n = 1 and X^3 + Y^3 = 1. These two problems are not the same. And anyway, if you admit Euler’s proof of FLT_3, then X^3 + Y^3 = 1 has no rational solution, hence the intersection will have no rational solution : a tautology. Which does not show at all, and will never show that X^n + Y^n = 1 in general has no rational solutions. This can be understood even by a school boy or girl. Stop wasting your time (and your money) with « elementary proofs » which have no sense.

To PSN (this should also interest Prof. Peter Breuer)

« I think that… », « The term condition is not… ». Enough of this repetitive ping-pong exchange where scientific discussion is replaced by unfounded assertions. Except at the beginning, I have never expected to rationally convince you anyway. And my free time is not infinite.

Before closing, I nevertheless insist to show you that your weird hypothesis (call it WH) that z, z^2 are irrational whereas z^3 is an integer is just wishing thinking. The problem is that I’ll need a minimal amount of field theory ( = the very beginning of Galois theory actually), which you can find in any textbook (for instance Serge Lang’s « Algebra »). So assume that z^3 = an integer m and consider the following chain of extensions (= overfields), using WH : Q = Q(z^3) < Q(z^2) = Q(m^{2/3})

My apologies to Mr PSN for my hasty conclusion against the WH. Actually my approach was correct, but my "performance" was to make two errors in the one line -1 (perhaps I shouldn't work at late hours!)

1) Apply the Eisenstein criterion (which is a sufficient condition) in a wrong way. My memory betrayed me: the sufficient condition of Eisenstein criterion requires the parameter m to be square free (the contrary of what I wrongly said)

2) The extension Q(z) / Q(z^2) is generally quadratic, but it could also be trivial, i.e. Q(z) = Q(z^2), as in the case described by Prof. Breuer.

One can manage to construct z in order to get what was announced, but this is not the general case.

In conclusion, WH could hold as well as non-WH. I wonder how Euler proceeded. This doesn't seem to be explained in Ribenboim's book, and the "explanations" of Mr PSN around WH are not clear to me. Concerning his "proof" of FLT, of course the situation is unchanged, as pointed out by many people.

P.N. Seetharaman

Preprint OBSERVATIONS ON THE OVERARCHING NATURE OF THE PYTHAGOREAN TH...

P.N.S.,

I think I am able to independently take care of the fate of my proof. In addition, one of the professors (you know this epic from the beginning of August 2021 to the end of September) was trying to announce\prove to me that I have no proof in my preprint 1 for the reason that I am not using the word proof when going from Scheme 2 to Conclusion. :) I have been explaining several times that I am perfectly familiar with the mores within the pure mathematics community and that I have been trying to avoid the word "proof" as much as possible. Even the title of Preprint 1 does not contain the word "proof". However, I used this word after the word "theorem" knowing standard rules for mathematical articles writing...no more.

In Preprint 2, I used the double term "proof/observation", although preprint 2 is most similar to "proof" in the literal sense. Then I was giving some hints, but so far no one pays any attention: I don't care whether Preprint 1 will be recognized as proof (without quotes). An attentive specialist can see something more in this preprint. I do not yet state here or in the "comments" field under the preprint - WHAT I mean. Because, I want to have verified attempts from people that have been listed by you.

I'm not going to get into "fierce arguments" with anyone here either, because this requires a new specialized thread. Such the thread was created ... Warning to the person who contributed to the closure of my branch: you are doing a useless and shameful action. You have no future.

Best regards,

Sergey Klykov

Here is the clarification by the president of the Clay Math. Institute

HYDERABAD‘Riemann Hypothesis’ remains open, clarifies math institute 📷Serish NanisettiHYDERABAD, JULY 01, 2021 00:01 ISTUPDATED: JULY 01, 2021 00:01 ISTSHARE ARTICLE

$1,000,000 on offer for solving ‘Millennium Prize problem’

“As far as I am concerned, the Riemann Hypothesis remains open,” said Martin Bridson, president of Clay Mathematics Institute, when asked about the claim by Hyderabad-based Kumar Eswaran of solving the problem that has puzzled mathematicians for past 162 years. Riemann Hypothesis is one of the Millennium Prize problems, for which $1,000,000 had been announced by the CMI from their inception in 2000. The problems are considered “important classic questions that have resisted solution over the years”.

The Riemann Hypothesis, postulated by German mathematician G.F.B. Riemann, is about prime numbers and their distribution. While the distribution does not follow any regular pattern, Riemann believed that the frequency of prime numbers is closely related to an equation called the Riemann Zeta function.“

I am surprised by the tone in which respectable publications in India are treating the claim that the Riemann Hypothesis has been proved. The speculation is rash and it would be wise to investigate more seriously about why leading journals and specialists in the field have not accepted this proposed proof,” said Mr. Bridson. Kumar Eswaran’s claim of solving the equation has been in the news since 2016. Mr. Eswaran, who is a faculty member at the Sreenidhi Institute of Science and Technology, could not be reached for comment. “I do not recall any contact from the author and I am sceptical about the merit of the review process that is alluded to in newspapers,” said Mr. Bridson, who said institute would be scrupulous in following the stated rules to evaluate claims that one of the Millennium Prizes has been solved.

On the website of Clay Mathematics Institute, the final word on Riemann Hypothesis is: “The problem is unsolved”.📷

Because of the "fuctional equation" governing the behaviour of the Riemann zeta function Z(s) under the change of variables permuting s and 1 - s (*), there is a so called "critical band" in the complex plane, defined by the real part of s comprised between 0 and 1. The vertical axis defined by real part of s = 1/2 plays a special role: RH states that all the "non trivial" zeroes of Z(s) are situated on this line. This is in accordance with all the existing numerical computations (by means of computers of course).

The connection between Z(s) and the distribution of primes lies in the expression of Z(s) when it is written as an "Euler product" of factors of the form (1 - 1/p^s) (of course in a certain domain of convergence). There are innumerable other applications in analytic number theory.

(*) In ANT, the complex variable is usually denoted s (not z).

@ Thong Nguyen Quang Do

I fully agree with your considerations, therefore I am attaching a my work , already published, which confirms that all the" non trivial "zeroes of Z (s) are situated on line s = 1/2" and that this is in accordance with all the existing numerical computation. (See "Table 1" and "Figure 2" on pages 134-135.)

Article AN ALTERNATIVE FORM OF THE FUNCTIONAL EQUATION FOR RIEMANN’S...

Obviously this is not a complete proof.

Sincerely

Andrea Ossicini

Who cares ? These are neither proofs, nor even mathematics, but mere divagations. The best reaction is to ignore them.

Peter Breuer ,

Thanks for the question. But, I was also writing above: "Because, I want to have verified attempts from people that have been listed by you (This answer was addressed to PNS, who mentioned me and I had to answer him.)." If you can, please help others with their proof. After that we could agree on a place in RG where I could give my answer or answers, if it will be possible.

However, I can say that I can offer more than one thing. [But, most likely, none of these things that were indicated in your guesses ...:-)]

Thong Nguyen Quang Do ,

"Who cares? These are neither proofs, nor even mathematics, but mere divagations. The best reaction is to ignore them." This is the best thing that can be offered to sleeping people ... No proof, no mathematics! Interestingly, what mathematics was proposed during the discussion of the last article by PNS? What was suggested IN GENERAL (if not to mention the articles)?

@P.N.S., *

I am very sorry that in a decent society we have to write about ugly things.

You are writing your comments, but then you delete their by dozens and hundreds. And after that you have such courage to make such statements?

"P.N. Seetharaman Seetharaman added an answer. 29 minutes ago

I didn't ask for any suggestions from anyone."

I mean, you call me a liar. Do you understand that? Or not?

Although, everyone knows who is the liar here and in another threads, in fact. It's good that I was doing some copies of your ... "messages ".

I was not a member of this thread, so I didn’t make copies here. I will give copies from another thread soon. Wait...

And stop writing all sorts of nonsense about me and others, because I'm tired from it already. Do you have something to discuss now? No. Then by what right do you accuse other people (me) of lying? Wait for copies ...

No respect ...

*P.S. ...to P.N.S.

I am making an attachment from another thread. Roughly the same thing happened in this thread. PNS, you are writing things that have not been asked of you. At the same time, you pronounce the names of people, including my name, for some purpose. Stop it, please!

And now you are the one who denies these facts. This is a complete lie ...

P.P.S.

:))) PNS, I think, after your today's idle talk, which we had now again seen from you, you have no right to expect something serious to discuss with you? Or am I liar?:)))

But, this time I had done my copies of your idle talk...

P.N. Seetharaman

The general equation has nothing to do with the cubic equation.

Both equations are quite distinct.

==================

This is exactly the problem....Not proven.

I already wrote to you: Okay, I am a silent one ... But it does not take even 2-3 days, as that every time you start your ...antics against me. I only had been forgetting about you, but you remind of yourself each time again and again.

Peter Breuer P.N. Seetharaman

For the reason that P.N.S. apologized to me (although he had these apologies deleted, but I had copies made for and by me ...) for his attacks on me and / or my materials, starting in April 2021, and also because P.N.S.would not like to see my questions from me to his proof (as I understood this, reading his yesterday's messages), I am in a stalemate.

I explain and I would like to ask: My questions/clarifications (for PNS) were very clear and without unnecessary words (at first, but after a while I started to insert metaphors like "Beatles songs"). Therefore, I do not understand what it means 2 of your remarks above:

1. "Please, both, do better" and

2. "But who knows until Sergey clarifies what HE means".

If you really have something incomprehensible, I could try to answer, for example, in a private form, so as not to annoy PNS.

+ As I had it written yesterday, I am not very free to speak in a sufficiently open form now, because the other applicants do not yet have any expert feedback on their attempts. And you did not answer this question of mine yesterday.

Could you say something about this?

Returning to the topic "proof from PNS", I can only say so far for more clarification: of course, the beginning of his article in the part of equations 1 and 2 is directly related to my proof / "proof" (you can use quotes or do not use, for me it is indifferent at this stage. But, I do not want to say that it will ALWAYS be indifferent to me.). It is obvious. That is why (apart from personal reasons) I had an active discussion of the very beginning of P.N.S. (Although, also, I have repeatedly checked his calculations. But for me it is less interesting and if I have done this work in the last 2 months to a very small extent, then only with the aim of finding specific inconsistencies or errors associated with equations 1 and / or 2). I wrote above that:

"PNS: The general equation has nothing to do with the cubic equation. Both equations are quite distinct.

==================

This is exactly the problem .... Not proven."

At the same time, I want to emphasize that if PNS had opposite statements in the article, such as" The general equation is equal to the cubic equation ", I could also find arguments "against" and they would have the same relation to my materials, as now.

That is, my main claim was and is that equations 1 and 2 (olf eqn 2)are ambiguous, therefore any "strong" statement that to give them some specific filling with the meaning available in the article of PNS cannot be successful ... P.N.S. have made equation (2) more specific now in his most recent version of the paper. It now has specific numbers in equation 2: 7^3+9^3=(4^2)*67.

Of course this is better. But, the problem of equation 1 has not disappeared anywhere. He writes "p is a prime number greater than 3". But why? Why should we believe that p is not an even number, for example? That is, if he eliminated the ambiguity in equation 2 by using a specific numerical example, then equation 1 leaves no choice for any statements that this equation is ambiguous.

Using the same aprroach, I will always be answering concerning my own PROOF/"proof": when someone asking me using words "p is prime", I will reply: "Why? I don't see the fact, that "p is prime", I see p=2q, where q is prime!" And conraversly, when"they" will tell/ask me "n (or p) is even", I will argue, proving, that "n (or p) is odd". And of course, we both can be right, because any "conclusions" like "p is..." will be far-fethched.

I guess I'll end there. My thought concerning next possible steps to improve of PNS's proof - it would be much better to try to obtain proof for FLT without using the controversial statement "p is a prime number greater than 3". For what purpose do you limit yourself? By limiting yourself, you limit your options.

Sincerely,

P.N. Seetharaman

Hence converting the general case equation into a cubic equation form is of no use.

===============================

I have not said/write this words anywhere. + This is a job for your brain.

@P.N.S.,

You have a few elementary mistakes, explained by your haste, in your comments above, in addition to "p is ...".

Please correct it. No need to rush somewhere!

I am leaving this thread for now, because I do not have something new to add to the messages that are in the RG from my side right now. If you have any questions, write to me via e-mail.

Dear Peter,

I would like to know why the argument of Krasner works only for large p.

No, you're mistaken, it works for all odd p. By the way, it also works against Mr PSN's latest "proof", but I wouldn't "dare" discussing with the author now, because he could always answer that he does'nt know what the p-adic numbers are.

Dear Peter,

In the Krasner trick, the chosen prime p is the exponent of the Fermat equation. And there is no proof theoretic argument at all. You simply must show that the equation x^p+y^p=1 has a non trivial solution in the field Q_p of p-adic numbers, which is the field of fractions of the ring Z_p of p-adic integers (the analogy with Q and Z is obvious). Just take the formal development as a power series of a p-th root of (1-Y^p) for p>2. If the indeterminate Y is given a value y in Q , the series is not convergent in R, which is the usual archimedean completion of Q. But in Q_p, which is the p-adic completion of Q, any series converges iff its general term tends to zero ! This phenomenon is due to the so called ultrametric inequality : denoting by /./ the p-adic absolute value, one has /a+b/

Yes, O1 seems to be sufficient to ensure convergece.

Bamboozled" = ? (even in p-adic terms)

Yes. Algebraic number theory nowadays systematically appeals to both the p -adic and the archimedean aspects together.

About Krasner's trick, it must be stressed that we work all along with variables x,y etc. taken in Q_p, the field of p-adic numbers, which is a complete metric space. Actually Q_p is the completion of Q w.r.t. the (ultra)metric p-adic distance, just as R is the completion of Q w.r.t. the usual archimedean distance. Algebraically speaking, one can similarly construct the ring Z_p of p-adic integers by completing Z. Topologically speaking, apart from their same mother Q, the two types are mutual total strangers: R is connected whereas Q_p is totally disconnected, so there is no risk to see solutions x, y etc. migrate from one space to another. One extraordinary phenomenon is described by the famous Ostrowski's theorem: up to equivalence (in an obvious sense), R and the Q_p's are the only complete fields derived from Q by the completion process above.

At this point, for a...complete clarification of the Krasner trick, recall that it is supposed to show us, without any calculation, why an "elementary solution" to FLT cannot exist. The point is to define precisely what elementary means: elementary arithmetic in Z is supposed to use only the natural calculations in Z , + and x, i.e. the ring structure. Some less elementary considerations sometimes appeal to the unique factorization, up to a sign, of integers as a product of (powers of) primes : this is baptized the factoriality property of Z , more briefly, Z is called a UFD (unique factorization domain). Elementary methods just mean that, and that alone. By construction, the ring Z_p is also a UFD, and much simpler than Z. Back to FLT, look at a pretended "elementary proof" in Z and transpose it in Z_p word for word, i.e. without any additional argument. If the pseudo-proof really worked in Z to show FLT, then it would litterally also work to show FLT in Z_p . But we know from the previous post that FLT_p is false in Z_p . QED

Dear Peter Breuer,

There are a few points which I would like to discuss in detail:

1) Why do you insit that , on the p-adic side, the variables x, y should be in Z_p ? The original Fermat equation is x^p + y^p = z^p, with x,y, z in Z. But its homogeneity allows us to divide by z^p, and consider instead the equation x^p + y^p = 1, where the new variables x, y are in Q. Same thing for the p-adic analogue with x, y in Q_p . This is only a detail, but it allows us to get rid of problems of integrality, and to work with rational variables (i.e. in Q or Q_p ) without caring much.

2) But perhaps you take care to stay in Z_p in order to be able to work modulo p^n ? There is no necessity to do so, and then to be obliged afterwards to come back to Z_p by taking the projective limit of the Z/p^n Z . Without this justification, I don't see how to "get a p-th root mod p^n". Surely this could be justified, but there is no need.

3) The last step, taking surds, rightly worries you. But this is because you made your life difficult when you chose to pass through the quotients mod p^n. Working directly in Q_p presents no problem at all.

4) Up to now, we can say we have been doing only p-adic analysis in Q_p to show that FLT_p is false in Q_p. The problem in discarding "elementary proofs" of the original FLT_p in Z or Q has not been tackled yet. Only a precise definition of "elementary methods" (as those we have witnessed so far in RG) allows to conclude that they cannot succeed. Because they can carried on in Z_p as well, word for word, they contradict the negative answer produced by the Krasner trick ./.

PS : you are right about induction, but I did not list it as an "elementary proof". Besides, induction on what ? On the degree of Fermat's equation ? Highly improbable, see the summary of the different proofs (e.g. to pass from exponent 3 to exponent 5) listed in Ribenboim's book. On the variables ? Impossible because Z_p is not an ordered ring.

For laymen in an area that has now been touched on for a few days, I am able to do a few posts so that laymen get more understanding. If you wish, can I continue?

Perhaps the professionals will move away from their language here and also help non-professionals? Nevertheless, this is my first comment not on my "traditional" topic:

There are very interesting entities in mathematics called "p-adic numbers". In fact, there is nothing complicated about them. However, in textbooks and encyclopedias, they are introduced in such a way that it is very difficult for the uninitiated to understand what they are talking about.

Here and now an attempt will be done to explain the p-adic numbers for amateurs.

To begin with, new mathematical objects should be introduced, conventionally called "quasi-infinite numbers", QIN, and some of their properties will be described/discussed.

And then it will be shown how to go from them to p-adic numbers.

So,yes or no?

@Peter Breuer

1) Since the ring Z_p is the inverse (=projective) limit of the quotient rings Z mod p^n, it has a more rigid structure than the mere collection of all these quotient rings because it encodes, and in a coherent way, all simultaneously information mod p^n for all n, which is remarkable, as you say. The field Q_p has no mystery, it's just the field of fractions of Z_p , just as Q is the field of fractions of Z. But Z_p has "no" mystery and wonderfully simple properties : not only it is a UFD (as Z), but it's a DVR (discrete valuation ring), i.e. more precisely, any element non null element z in Z_p can be written uniquely as z=up^n, where u is a unit (= invertible) and n is in N. The same type of decomposition is valid in Q_p, with n in Z. Can you imagine a more simple ring/field ? André Weil used to say (I think): "In the 6 first days of Creation, God introduced the Q_p 's. Then, the 7th day, the Devil came and invented R !" .

4) The DVR property of Z_p is much simpler than Euclidian division. But at the same time, it prevents us to use the traditional "descent" trick (remainder < divisor, etc.) because Z_p is not an ordered ring. Same objection to 5).

No, "QIN" is a must (to have for non-professionals to make it easier for understanding).

"QIN"s are written in order from left to right:

"Quasi-infinite number" ("QIN") is an infinite sequence of numbers (from any number system, such as decimal), going from right to left.

Example: ... 2708137282573017720328173467

These numbers are called "quasi-infinite numbers" ("QIN") because they appear to be infinite, but in reality they are not.

Looking ahead, I want to say here that for different number systems (if I just had it expressed correctly in English now or is the term "radix" more correct?)

It is necessary to talk about such a fact as "Non-equivalence of different number systems" (radix?) *.

That is, in the decimal system, you can represent 1/3 (like ... 66667), but you cannot represent 1/2 - because when multiplied by 2, the last digit always turns out to be even, and we should get one. In the ternary system, on the contrary, you can represent 1/2 (as ... 11112), but you cannot represent 1/3 (when multiplied by 3 = 103, the last digit always turns out to be 0, but one should turn out).

Thus, it can be noted that in this number system, only those (irreducible) fractions in which the denominator is mutually prime with the base of the number system can be represented in the form of "QIN". Therefore, "QIN" written in one number system may not have any correspondence in another number system.

But, we have not yet moved on to p-adic numbers yet. But, this transition is simple. They almost do not differ from the above-described "QIN" , however, they have the following features:

-The base of the number system is always a prime number.

-The numbers are written in the reverse order compared to the above (that is, the endless tail goes to the right, not to the left; however, this is only a form of notation, the essence does not change from this).

-The numbers themselves are called "p-adic numbers".

I had obtained this information through Internet-consultations with Dr. A.V. Lukyanov. Although when I was a student we were studied this very quickly and I hardly remember. Honestly speaking, I am almost remeber nothing about this subject. At another faculty (cybernetics of chemical technological processes), this should be studied in more detail, as I understand it.

However, specialists in this matter have a much greater understanding and it would be better if this information was given at an elementary level from the very beginning in the way I am doing now with this second post of mine. If there is a need, in the third commentary I could give information about elementary primitive operations with QIN, about integers and negative numbers, fractions, square roots, complex numbers, "QIN" modules.

However, I see no interest from non-professionals and I will think more than once, before I will be writing here on a topic that is unusual for me.

*P.S. This can be a key factor when considering FLT, right?

@ Peter Breuer Things are simpler than you could think. The origin of the puzzling "paradoxes" that you bring out is that, although Z is contained in Z_p (and Q in Q_p), you must beware that Z or Q can be non stable under certain p-adic manipulations: as I stressed, same mother, but different breeds. Lett me center my recap around a theorem that I cited in my post of 1 day ago: every non null element x in Z_p can be written uniquely as a =u.p^n, where u is a unit (i.e. invertible) of Z_p and n is in N (*).

First recall a few algebraic considerations . A naive definition of a p-adic integer a in Z_p (where p is a fixed prime) is a = a sequence of integers in Z, {a_0, a_1,…, a_n,…} s.t. a_(n-1) is congruent to a_n mod p^n for all n > 0. This is just the elementary translation of the concise definition Z_p = proj. lim Z/p^n Z (as rings), where the connecting homomorphisms are the natural ones sending a mod p^n to a mod p^(n-1). The proof of (*) is not much more involved, it takes at most one page of "elementary" calculations. Wahay! (= ?) Let us explain incidently why Wiki. hints at "the completion of the localization of the integers at the prime ideal generated by p" (1 day ago). We can see in passing that an integer a in Z, considered as a p-adic integer in Z_p, is a p-adic unit iff a is not a multiple of p. And so, any rational number of the form b/a (with b in Z) will be in Z_p. Such rationals are called p-integers, which constitute the localization of Z at p.

Now some p-adic analysis. The usual algebraic construction of the field of fractions of a ring which has no divisor of zero (= a domain) gives us Q_p .

Changing notations if necessary, we can at once generalize (*) to Q_p : every non null element a in Q_p can be written uniquely as a = u.p^m, where u is a unit (i.e. invertible) of Z_p and n is in Z (**). Notation : the exponent m of is called the p-adic valuation of a, written m= v(a) (we drop the index p for convenience).Then the definition of the p-adic convergence and (**) show that any non null p-adic number a in Q_p can be uniquely written as the (convergent) sum of a series a = p^m (a_0 + p a_1 +…+ p^n a_n…), with m=v(a), 1

I have already written that this is used to make it easier to understand. There will be no fundamental objections from my side, if this term is dropped and not used. However, it will be difficult to explain (at least for me) when a given entity has no name, although this name is even given in quotation marks.

Plus, I am following a form that is not my "property" - I am now as a kind of repeater. Ie, please, all questions should not be addressed to me ...

When we go over to the p-adic solenoid, it will be easier to understand if we have the original term "QIN".For example.

However, I see no interest from amateurs. Let the experts give explanations, but I'm stopping my participation right now. It would be more better and useful to read explanationd from these experts. But, please, use more simple language for this...And it would be good thing, if we can see examples, please.

I am not debating a topic in which I am not an expert. In my opinion, I said this more than once above, although in other words. I am just taking the first steps, as a "retranslator", so that other applicants for the FL "theorem" are not afraid of any names, titles, categories. They must read more. Including simple sources with understandable explanations.

Therefore, once again: all objections are not to me! At the same time, it does not matter at all what it may be, p-adic numbers / solenoid, "QIN" (regardless of the correct or non-correct using in this context) and others. In the end, the concept of "infinity" even here (in the RG) is interpreted broadly in the sense of recognition of this essence, to complete denial. It is a matter of the narrator's "taste" and experience.

The main thing is different, I wrote it above, but everyone else should understand it deeper and use, I mean this:

"" Non-equivalence of different number systems "(radix?) *.

That is, in the decimal system, you can represent 1/3 (like ... 66667), but you cannot represent 1/2 - because when multiplied by 2, the last digit always turns out to be even, and we should get one. In the ternary system, on the contrary, you can represent 1/2 (as ... 11112), but you cannot represent 1/3 (when multiplied by 3 = 103, the last digit always turns out to be 0, but one should turn out) ".

In turn, this understanding makes it much easier to get an answer to the question in this Discussion:

https://www.researchgate.net/post/Do-irrational-numbers-exist-in-nature

Therefore, once again, there is no interest, there is no point in wasting my time here. + Should I receive veiled or open reproaches here for my underdevelopment in comparison with the "elite" , everyone understands who calls himself that way. No, I do not need it. If someone wants, we can limit ourselves to private correspondence ...

But, this does not mean that I will never return here or anywhere else with the traditional topic of FL "theorem".

"If you are talking about 10-adic INTEGERS ..."

============================================

Yes, if you want. But, do not knock me off the line that I started a couple of days ago. QIN in the decimal system...

Square roots.

Example. In the decimal system, it is easy to extract the square root of ... 00004 - these will be ... 00002 and ... 99998 (it is clear that square roots always appear in pairs).

Theorem. If the base N is a prime number greater than 2, then for any number x not ending in/with 0, there is a square root √x, provided that there is a number y0, which, when squared modulo N, gives a number equal to the last digit x.

Proof. The last digit of the root y0 is known from the condition. Let's choose the next digit of the root yi.

Based on the column multiplication algorithm

2 yi y0 + C = xi, (I didn't show it here because I was forced to jump over to the question asked about "square roots"; calculations are carried out modulo N, xi is the next digit of the original number; C is the "appendage" resulting from the multiplication of the previous digits).

Since y0 ≠ 0 and N is a prime number greater than 2, this equation is always solvable. The theorem is proved.

QED.

Example 1. In the 7-ary number system √2 = ... 266421216213 and ... 400245450454 (here it is no longer possible to determine which root is positive and which is negative).

Example 2. In the 7-ary number system, √3 cannot be written, since no integer when squared mod 7 gives 3 (that is, you cannot find the last digit).

Example 3. In the 11-ary number system √3 = ... 761192486 and ... 349918625.

My apologies for the 5-th time, probably, but it would be better to get such information from experts, right? ........

1. "the decimal system"-Need more clarification.*

2. and 3: You, apparently, also have not understood me correctly (because of my English?) - this is me, who is waiting for clarification on this matter...

But, okay. I think you could continue on about p-adic numbers without me.

For me, the answer that was obtained upon a cursory acquaintance with this question is much more important: FL"T" is solvable only in quite certain conditions regarding "numbers systems"! And when I was writing "we are working within the framework of 17th century algebra"- I was / am not only right, I am absolutely right. The problem is in numbers systems.

Despite the" additions "and other very important logical changes. We are now even we are not talking about non-Euclidean geometry, where the sum of the angles of the triangle is not equal to 180 degrees - we are talking about another level - the possibility or impossibility of the existence of specific numbers under certain conditions(numbers systems)! This is the most important thing.

Regarding the possibilities of further improving my qualifications for a p-adic, I only welcome this! I will do it gradually, step by step. And for this reason, I welcome any help, any participation in this help for me.

*P.S. Sorry! :^) Understood! :^) Thank you.

I.e., one might possibly and inconsistently choose - and + square roots mod pn and mod p^(n+1)

--------------------

I had given specific examples.

These examples are obvious to make the final conclusion that I was looking for my understanding: the number system for the p-adic is the determining factor to be making judgments about the FL "theorem". This is enough for the Question here (above).

The rest of the questions about the p-adic system are interesting and certainly deserve attention.

That's all for now.

"obvious" is not a proof, "examples" are not a proof

======================================

When you are reading in some mathematical or other text √2, you do not prove it, but accept it AS IS, if it is not required of you by some special conditions.

Likewise, you was reading example above "In the 7-ary number system √2 = ... 266421216213 and ... 400245450454". Here, too, there are no special requirements and conditions for me and for you. You should just be doing the appropriate multiplication √2 * √2 to make sure it is true or wrong.

You are convinced that the error does not exist in this case.

If you look at the other examples (above), you are well aware that some numbers do not exist at all- for DEFINITE number systems. [Example 2. In the 7-ary number system, √3 cannot be written, since no integer when squared mod 7 gives 3 (that is, you cannot find the last digit).] Is it TRUE or FALSE?

That's all. What more "proof" is needed?

Several times I had it written "please discuss without me". Because, I had made the main conclusion for myself. FL "T" is more interesting for me and it was for FL "T" that I had a little time spent on p-adic. A very interesting thing! I'll watch/learn/study some more, but a little later.

Good luck.

KSP: "some numbers do not exist at all- for DEFINITE number systems." Is it the most important thing in the context of the topic that is currently under Discussion? I hope I am understanding this topic correctly. If you have such a strong desire not to see this theorem more, that has caught your attention, I am in a position to edit or to delete this comment of mine above. Yes?...

Yes. There is no solution...in the..., and no solution to...in the..., and there is no solution... in the...

========================================================

Yes, no numbers. Integers, rationals or even "irrationals"? Not "?", but only "!". Hmm. But, these numbers exist in other systems ...Why are there so many words?

Good luck.

P.S. √3=no solutions, hence, 3=(no solutions)2. Yes?

Perhaps this is another language thing.

==================================

Dear Doctor,

I don't think the main problem is with "language". Despite the fact that you are writing to me “I don’t know (what it is ...)” or “I don’t understand (what it is)”, I am sure that you understand correctly.

For example:

1. If I am writing about the "system of numbers", then I mean the way they are written // represent. For example, if we have 5 in "decimal", 510, then it is easily represented as 101 (one-zero-one, not one hundred and one ...) in binary. That is, 510 = 1012; 510 = 123; 510 = 57; 310 = 37 .

2. If I am a writer about p-adik numbers, I mean the representation of these numbers according to paragraph 1 in any system.

Considering paragraphs 1 and 2, there is no point in talking about the absence of a language in order to understand my yesterday's statement that the septenary system of calculus does not give sqrt(3).

I think the example with "the amount of money that does not exist" is not relevant, because the amount of money that IS is always known - the Federal Reserve System (or the European Central Bank, or London financiers) always knows the amount of cash and the amount of non-cash money. Therefore, taking into account the concept of "infinity", we can always find "the amount of money that does not exist" by means of an elementary arithmetic operation - by subtracting a known amount of money from infinity. : ^) (Joke).

Returning to the "number systems", I would now like to give some of my observations as the main idea / question of this commentary of mine.

It will sound like this. If we "have" irrational numbers sqrt(2), sqrt(3), etc. in the "decimal system", it is logical to assume the existence of analogs in any other "number systems". Now we are not talking about the real possibilities of the existence or non-existence of these numbers ... We will assume that these numbers exist.

However, I came across the fact that the number "7" represents a certain peculiarity in this question in comparison with other numbers - see. the same example sqrt(3) in the septenary system ...

Nevertheless, my thesis about the non-equivalence of number systems remains valid. Therefore, any judgments about FL"T" from the standpoint of different number systems should be made with great care.

Regards,

Dear Peter Breuer,

While you're at it, would you mind giving a look at the appended (unsollicited) invitation which I have just received ? Fun is guaranteed. Best, Thong NQD

In view of the childish math. level of these "extremist amateurs" (Carlos de Matos is a lawyer, and a constitutionalist at that !), you can always hope but be prepared to wait !

@ Thong

There are imitators to every serious mathematicians .And they get more applause from the non- mathematicians , and they want to bring maths to their level. In India there are many crooked people who claim themselves as Messiah or World teachers , and they survive because they are able to attract by cheap means Mr

Sarva Jagannatha Reddy has lot of people who thinks he is another Ramanujan , because there are people who bring Mathematics to the low ebb

You are looking too far. The unique factorization in Z_p comes from the property that I denoted as (*) in my post of 2 days ago : every non null element a in Z_p can be written uniquely as a =u.p^n, where u is a unit (i.e. invertible) of Z_p and n is in N . This can be immediately checked, recalling that "uniqueness" means "up to units". The simplicity of the phenomenon is easy to explain: there is only one prime, which is p itself ! It's a kind of "super-localization" at p. By the way, you can't speak (without confusion) of "n-adic" integers or numbers when n is not a prime because of (*). True, if n is a multiple of a prime p, you still can speak of n-adic expansions, but the "ghost in the machine" is p, any other prime q becoming automatically invertible. That is why "arithmetic in Z_p is not interesting in itself, it's the whole universe of the p-adic (for all p), confronted with the archimedean universe of R, which produces the new sparkle.

I take this opportunity to come back to the distinction you made between the philosophical approaches of Socrates and Plato a few days ago. I have never fully accepted the philosophic "cliché" which people use to "explain" the "ureasonable efficiency of the mathematical language " in detecting the order (or symmetry, or harmony, or whatever) hidden behind the laws of Nature. On the contrary, one could say that the discovery (or invention) of such laws, because they are more deeply hidden, should rather reveal the pre-existence of the idea (in the platonist sense) of symmetry behind the disorder of reality. The italics everywhere are warnings that here we are on the borderline of an endless metaphysical (not to say tautological) discussion on the notion of reality. Speaking of platonist concepts, let me stress that they seem to me historically dated, corresponding to a certain - primitive - stage of scientific knowledge. Nowadays one could as well lean on the discoveries in neuro-sciences to assert that Plato's ideal world is just the world of our brain, which is the indispensable intermediary in our perception of reality. Then, no wonder when we recognize patterns / structures in our description of this reality since it is the same organ which organizes (regurgitates) what is has perceived (ingurgitated). Surely metaphysicists would howl at such a materialist interpretation. But to stay with the platonist imagery, I would say that we discover mountains and streams, but that we build (invent) bridges and roads to come across.

Back to mathematics. One example being better than one hundred speeches, as the French say, let me give an example which I think is the perfect illustration of what we are discussing about. The starting point is the famous Bloch-Kato conjectures on the "special values" of the Zeta function (or more generally, of the L-functions), which generalize Dedekind's "analytical class number formula" and reveal a deeply mysterious relationship between the transcendental world (the above functions) and the algebraic world (the ideal class group and the higher K-groups in number theory and algebraic topology). These conjectures were proved only two decades ago, at least for abelian number fields, thanks to original - not to say platonist - ideas of Grothendieck, the so-called quest for motives (in the sense of the impressionist painters).

This story strikingly illustrates Plato's "apologue of the cave": we live in a cave, and the physical reality which we perceive consists in shadows cast on the walls by the sun in our backs; to understand the true reality, we must turn around to face the archetype which projects these shadows. Grothendieck applied this philosophical concept to algebraic/arithmetic geometry : around a given geometric variety are floating a host of dissimilar cohomologies (Betti, de Rahm, étale...), which become isomorphic when passing to an algebraic closure of the base field, but such a passage makes us blind to all the arithmetical properties we are interested in. Following Plato, Grothendieck suggested to look not at the shadows but at the archetype, the conjectural "motivic cohomology"... A harebrained idea, it may seem, but which has occupied people for decades, until the final success of Voevodsky (Fields medalist in 2002). Vv. immediately used his new weapon to bring down two other conjectures (by Bloch-Kato and Lichtenbaum-Quillen) giving precise relationship beween K-theory and Galois/étale cohomology, and that was how the conjecture on special values of Zeta was settled. So, as a mathemathician, are you a Platonist or not ?

Anything (commonly) known then? No, because it would contradict Ostrowski's theorem (4 days ago) : One extraordinary phenomenon is described by the famous Ostrowski's theorem: up to equivalence (in an obvious sense), R and the Q_p's are the only complete fields derived from Q by the completion process above.

@Peter Breuer

Regarding 2 points discussed above with/at disagreement:

1. "QIN"- Russian-language literature on this topic has this to understand.

2. "... Not only does it not" remain "valid, it was never" valid "in the first place, being meaningless" ... Not at all. It makes no sense to answer with many letters to my simple example of the absence of a square root 37 in a 7-adic system ...

Etc.

Therefore, for items 1 and 2, you can write not to me more. Because I am not able to change the Russian-language literature in point 1;

I suggest that you discuss this example in point 2 not with me, but with someone else.

Good luck.

It was many words again for my simple question/statement.

Any one can read above, that I was writing my question not for experts. Okay.

Are you a teacher? Good.

No, it is more correct to say the Teacher. New Euler? No, no ... Probably, new Newton? Joke. Joke?!:))

One cannot fail to note the skillful mastery of polemical methods. This was manifested almost immediately after the appearance of the topic of p-adic numbers here, when I bluntly said that I am not an expert in this area. In this case, what is the purpose of the understanding/misunderstanding questions? (But, the subject is not hard, it is possible to get not very bad understanding for a short time.)

For what purpose are questions about π? A number that has nothing to do with the context that I started a few days ago ... Probably to check common knowledge from "student"? And to continue position of Teacher. To shame student? ..

Vaccines? Why is it here? Are these questions for me as person with much more experience in this matter? Hmm ...

Are vaccines connected directly with "p-adic?" :) Or with π? Why are there many issues out of my simple questions?

There are reproaches, because of references to "authorities", but is Wikipedia as an authority for Teacher?

Anyone can see above that, when I was starting the topic on this numbers here, I wanted to discuss some simple things. I didn't and don't need polemics. That was done for not experts. And what can we see now? Teacher is trying to drag me into this unnecessary controversy, instead of giving ELEMENTARY answers to ELEMENTARY questions. I don’t have as much time as Teacher to write very long comments many times per day with links to very authoritative sources like Wikipedia, unfortunately.

So, no answer on my issue until now. Okay. I already don't need it now. I hope you had understood this right now.

Nobody had given a simple answer to my simple question/statement.

But, you should understand good, that issue is contained not in one example*. There are other examples. Yes?..

Not having a simple answer can only mean 2 things:

1. The question is really stupid and this stupidity can be easily demonstrated using certain elementary means. These means a normal teacher is a giver with the help of a language that is accessible to the questioner. If a teacher does not know how to speak ALL languages, such a specialist loses the right to have the name "teacher". (Of course, in this case we are even not talking about "Teacher" at all!)

2. The question has the right to exist in any human community. (I have a suspicion that my question belongs precisely to group 2.) In this case, the "teacher" begins to show dishonesty and other negative traits when, instead of a sane explanation according to paragraph 1 above, he begins to speak wordy and to be trying to look like an EXCELLENT polemicist to others ...

That is, someone is preoccupied with his own image or the image of something else. Of course, someone can consider me an idiot, but everyone understands perfectly well whose / what image the polemicist above additionally cares about.

The reference to Krasner's trick betrays the real concern of the polemicist ...Also, it is the essence included in the FLT and things, which are close to this, including maths experts opinions.

I am responsing: you are right in this matter. The only thing I want is to get more understanding.

I wanted to offer you some Exercise for a long time. This is an elementary task. Once you answered “I don’t understand, you don’t know the language”. This was in a private discussion. However, the task is elementary and I would like to repeat my question publicly: could you try to solve this problem in order to help me, or do you give your refusal again (this time, publicly)? If you agree, I am ready to publish the terms of this Exercise here.

* P.S. Of course, the search for other examples will continue if I have time for this. Whether it is with the help of RG here or aside from RG does not matter.

"...Find some defining equation for π, for example, and prove there is no p-adic number that solves it?.."

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Perhaps this tip of mine below can play a key role.

Or no.

As I have already said many times, I did not have such a task before me. But, if I have received such a request from you, I am able to offer you my hypothesis*. In order not to search in mathematical reference books, I do not have much time now. Therefore, it is easier for me to put forward a hypothesis, but someone may try to prove or disprove it.

My "hypothesis" is presented below:) :

π/6=Φ2/5, where Φ is the major "golden ratio", hence, π/3=3/5+[sqrt(5)]-1.

Or consider it as: π/6≠Φ2/5, hence, π/3≠3/5+[sqrt(5)]-1 .

Where is true?

I will not go into details - why I am proposing such a solution.

Take this as a basis and you can try to refute or to prove.

You will have any integer that can be expressed in terms of Φ, in the case of proving the truth of this relation.

(At the same time, we remember about the "Basel problem" with π2/6, but we are not talking about that now. However, perhaps you will be able to sum the proof or refutation specifically under the "Basel problem", I do not know ...)

Good luck.

*P.S. I suspect things might be wrong here, but I haven’t refuted it yet. Perhaps someone will do it easily and quickly.