One problem with the proliferation of "open access" journals is the decrease in quality. A good example is this "proof" of Fermat's Last Theorem by a guy who seems to specialize in rather eccentric papers. This paper was passed around to great laughter at the van der Poorten memorial conference in Australia. (The list of keywords alone is funny to a professional mathematician.)
This journal - the Journal of Mathematical and Computational Science - and its editorial board should be ashamed of publishing this junk.
Sunday, March 18, 2012
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I had a long correspondence with one of these folks. A fine retired gentleman who made one mistake after another in an attempt to use basic high-school math to prove FLT. I kept telling him that although I would never say never, the tools that he was using were simply of too feeble mathematical content to be likely to prove anything. He persisted till I gave up answering his e-mails.
Wasn't this proved by a mathematician some time in the 1990s?
Should I infer that you are opposed to the basic idea of an open-access journal, or are you simply pointing out that open-access can lead to a proliferation of bad ones? (I happen to believe that open-access journals are a great idea, but they place ever more responsibility for integrity on the editors.)
SLC:
Yes, Fermat's last theorem was proved by Andrew Wiles (with some help from Richard Taylor).
That's the problem with false proofs of true theorems: it's not easy to produce a counterexample.
isohedral:
I edit a true open-access journal - that is, free to both authors and readers - so evidently no, I am not opposed to them.
There are a number of problems with many open access journals as they are currently set up. One is that most such journals have no provisions for waiving the author fees for those without a grant or those in 3rd world countries where the fee to publish might represent an average year's salary.
Thanks for the amusing link. (Surely he is credible, look at how diligent his research review was, why, he asks us to "see [1] or [3] or [4] or [5] or [6] or [7] or [8] or [9] or [10] or [11] or [12] or [13] or [15] or [18] or [19] or [20] or [21]"!)
However, I don't know why you focused on the openness of the journal's access. Closed journals are just as capable of publishing junk. Just look at Chaos, Solitons & Fractals.
Journals that make their money through author charges, rather than through institutional subscriptions, have less reason to insure quality, and more reason to accept nearly everything submitted.
Indeed, one might properly regard the Chaos, Solitons and Fractals issue as also being caused by a departure from sane journal-funding strategies.
If Elsevier sold journals individually, loads of people would want Advances in Mathematics (for example) and pretty much nobody would want Chaos, Solitons and Fractals, and it would not seem like such a big deal at all.
Apparently the author was charged $100 for publishing this paper. Forty papers have been published so far, and $4000 is a decent profit for putting up a web site. (It's clear from this paper that the level of copy-editing provided is minimal, if it's nonzero at all.)
Sadly, I recognize several names on the editorial board, probably people who have not closely looked into the standards of this journal:
http://scik.org/index.php/jmcs/announcement/view/5
Strangely, I know about one third of the editors... I am not sure they are aware of what is going on.
I didn't realize your journal was open access. I'd be interested in chatting with you about that at some point, particularly since I just took over as editor of a non open access journal in January. Do you have a web page up that explains the mechanics (policies, payments, copyediting, etc) for the journal?
Iso:
There are no payments! As I explained, it is completely free to both authors and readers.
You can see information about copyediting on the Journal's home page: http://www.cs.uwaterloo.ca/journals/JIS/ .
The paper is ridiculous. Thanks for the laugh. I shared it with my colleagues in the math department at Uppsala today and also linked it on my blog.
Incidentally, please be informed that the Goldbach conjecture has now been proven to hold:
Ikorong Anouk Gilbert Nemron. A proof of the Goldbach conjecture and the strong attachment to the Fermat's last conjecture. South Asian Journal of Mathematics, Vol. 1(2), 68- 80, 2011.
I note that Ikorong Anouk Gilbert Nemron is part of the editorial board of the South Asian Journal of Mathematics.
Nemron is not the only editor of the South Asian Journal of Mathematics who has published a "proof" of the Goldbach conjecture in the South Asian Journal of Mathematics:
http://www.sajm.com.nu/sajm2012_2_2_16gandhi.pdf
A Google search for "Ikorong Anouk Gilbert Nemron" yields several papers more or less identical to his FLT paper "proving" other famous results.
Is Nemron actually a member of the Centre de Calcul de Enseignement et de Recherche of the Université de Paris VI (Pierre et Marie Curie)?
The South Asian Journal of Mathematics seems to be full of weird papers, such as http://www.sajm.com.nu/sajm2012_2_2_11gueye.pdf. Aside from what must be translation issues (switching between "prime" and "first"), some things just seem inexplicable. For example, n^2 is used to denote squaring, but n^* is used for cubing.
Well, his address is ...@ccr.jussieu.fr. Replacing this by www.ccr.jussieu.fr brings us to web page of a department which looks like computing support division. In other words, he's probably a computing officer (centre du calcul, in french, probably means computing [support]). So, my guess is, he's a computer hacker with lots of free time and/or good copy/paste skills.
"A good mathematical joke is better, and better mathematics, than a dozen mediocre papers." - J. E. Littlewood [taken from the quotes page of UC Berkeley Prof. Emeritus Paul Chernoff]
In this case the joke and the (less than less of a) mediocre paper are identical.
See the note on http://people.cs.uchicago.edu/~razborov/.
See Wile's Proof of Fermat's Last Theorem at http://www.coolissdues.com/mathematics/Wile'sproofofFLT.html
If you find Andrew Wiles proof is difficult, Pl. read on the intenet(open access pdf)CMNSEM,Vol.2 ,No.3,March 2011(General proof),CMNSEM,Vol.1,No.6,September2010(correct and simple use of method of infinite desscent of Fermat, in Case.N=3)CMNSEM,Vol.Vol.1,No.3,April 2010(except trivial typos,wonderful proofs seem OK)Extremely short simple proof for n=7 in CMNSEM,VOl.1,No.9,December 2011,However, a mistake may be seen in n=5 case published in the same journal.
Simple proofs for specific exponents have long been known and aren't of that much interest.
I had never heard of CMNSEM before. It is apparently the Canadian Journal on Computing in Mathematics, Natural Sciences, Engineering and Medicine. It does not appear to have very much of interest in it.
The journal lists no editorial board and the accompanying web pages are written in ungrammatical English. Not very impressive.
Thanks a lot for valuable information on open acesss journals and Fermat's last theorem.The Author,Ikrong Anouk Gilbert Nemrong has already proved Fermat's last theorem seperately-A Complete Simple Proof of the Fermat's last theorem,International Mathematical Porum,Vol.7,2012,np.20,953-971. The paer (wonderful one,if it is OK) in the Joiral of mathematical and computer science had been reviewed within ten days!
Ikorong Anouk Gilbert Nemron has already proved Dermat's last theorem seperately.(A Complete Simple Proof of Fermat's Last theorem,International mathematical Forum,Vol.7,2012,no.20,953-971),This is not Mathematics,I guess,but Logic,etc.. I don't understand.Another one.A Simple Proof and Short Proof of Fermat's last Theirem,Fayez Fok Al Adel,Advances in Applied Mathmatical Analysiss,Vol.3,No.1 (2008)7-15,I never seen any comment on this simple proof,Even though author asked to comment on this proof seperately.This year I did not see it on the internet,I don't know why. Journal is open or not , the correct proofs must be appreciated and recimmended by people like you since mathematically sound Andrew Wiles-Taylor proof is extremely difficult and lengthy. Fermat's marvelous proof(?) is also available noW. Author is a professor.Daniele De Pedis Anyway, your information is useful to Honest Researchers. Thank you.
The Author,Ikrong Anouk Gilbert Nemrong has already proved Fermat's last theorem seperately
No, he hasn't.
It is, A proof of the Goldbach conjecture and the strong attachment to the Fermat's last conjecture by Ikrong....NEMRON ,the paper on the South Asian Journal of Mathematics, 2011,Vol.1,No.2,68-80 had been reviwed within ten daye. I am soory, it is not the journal of mathematical and computational science.Thanks to you , anyway, we know that all big problems in mathematics have solved!!.Thank you also for your cooment on the same person's proof of Fermat's last theorem.
An Anonymous (not me) has reported the notice about the pubblication of my demostration of Fermat's Last Theorem.
If you are interested in, you can found it on
http://arxiv.org/ftp/arxiv/papers/1105/1105.0669.pdf
Any comment/suggestion are welcome
daniele.depedis@roma1.infn.it
It is true that empty vessels make most noise.
Publishing of nonsense in the open access journals seem to be possible then and has been done as well.This acctually kill the moral of honest reserachers who do not want fame or anything.There are many valuable results in open access journals which we can study freely.This must be apprected while hackneying nonsense,
See BealFermatPythagorasTriplets.htm
I'm sorry to come rather late to this discussion, but the "South Asian Journal of Mathematics" has just been drawn to my attention via another route, and a web search for that journal led here.
Over the last few years various vested interests have promoted the "author pays" model of publication under the weaselly phrase "open access" (or even "gold open access"). There have been warnings that the adoption of such a model would lead to a rash of poor papers being published in fly-by-night journals. What I have only now realised is that this is already happening.
As regards I. A. G. Nemron, looking at his abstracts for Concerning the Goldbach Conjecture and Fermat Last Assertion in Commmunications in Mathematics and Applications and A Curious Strong Resemblance between the Goldbach Conjecture and Fermat Last Assertion in Journal of informatics and Mathematical Sciences reveals them (the abstracts) to be virtually identical. Whether the papers are identical, I cannot comment, as both journals adhere to both the doctrines "author pays" and "readers pay".
The elementary "proof" posted on arxiv by Daniele De Pedis is flawed.
At a certain point he claims that gcd(A, B, C) = 1 implies that gcd(B+A, C-A)=1.
This is clearly false. A counterexample is:
A = 2
B = 13
C = 37
Thanks a lot to the genteman's comment on Daniell De pedis elementary proof.
I think no researcher will be interested in commenting on the so called elementary proofs of Fermat;s last theorem not given in refereed journals.It is utter waste of time.Many of these authors do not know Fermat riples do not satisfy Pythagoras' equation.This is, by no means a prof of Fermat's last theorem.Unfortunately, some of these proofs have been given by Professors.
Prof.Fayez Fok Al Adeh's proof of Fermat's last theorem is not simple at all to me,a university teacher, which has been published in the Journal of Advances in Applied Mathematucal Analysis. However,I am trying to understand it.
Answer to Paolo Tassotti.
Your counterexample is not valid, of course the A, B and C variables must fulfill the 's equation!
On the other hand, the property gcd(B+A, C-A)=1 is reported also in the P.Ribbemboin book- "Fermat's last theorem for amateur".
Clearly the statement must be interpreted as: "if there were three integers that satisfy the Fermat's equation then ..."
In any case, recently I have posted on http://arxiv.org/pdf/1105.0669.pdf a new version of my proposed "proof" much more simpler and general, any comment and/or suggestion are welcome.
My request message to the moderator.
On 12:21 PM, August 13, 2012 I have posted my answer to Paolo Tassotti concerning my "proof" of the Last Theorem. In that post,due to my inexperience about this blog, appear two errors:
1) The author of post is Master instead of my name "Daniele De Pedis"
2)on the first row is missing the name, that is:"...must fulfill the 's equation!" instead of "...must fulfill the 's equation!"
Please could you fix these two points ??
Thank for your help
Daniele De Pedis
By reading this Blog(very very yseful), I understand that CMNSEM(Canadian Journal of Computing in mathematics,Natural Science,Engineering and Msrdicine) does not contain the general proof of Fermat's last theorem. This is rather unfortunate.Simple and short analytical proof of Fermat's lasttheorem is there.This will certainly help to reduce the calculation of Prof.Daniele De Pedis(Wrong!) proof drastically.Would you please point out a simple proof of Fermat's last theorem for n=3 case that I( University firstyear student of a third world country) can understand.Thank you very much regarding your comment on CMNSEM as well.
Take it easy, whatever you have heard or read about CMNSEM.It has already published 'A simple and snort Analytical proof of Fermat's last theorem'.CMNSEM is a refereed journal.
It has already published 'A simple and snort Analytical proof of Fermat's last theorem'
I rest my case.
I am going to killed.Long live CNMSEM!!
Thank you so much.Don't you like to comment on the validity of the proof in CMNSEM,Any way ,thank you so mych.Thanks to God ,you are a proffesor in Canada.
Let me join this discussion since I'm also engaged in despicable business of proving FLT. Here is an address of the abstract of the offered proof. Just in case if it may interest somebody
http://www.fileden.com/files/2012/5/4/3300464/Abstract%20of%20short%20Proof%20PDF%281%29.pdf
I am the euthor of A simple and short analytical proof of Fermat's last theorem in CNMSEM and some other as well.
Thanks to all.
I don't argue with the people. Howwever,proving Pythagorean triples do not satisdy the Fermat equation(General)eqyation is not the proof of Fermat's last theorem.
No one can nullify the papers I have published in CNMSEM.
Answer to Anonymous on 2:28 PM, August 13, 2012
If you have found any error in my proof posted on arXiv.org please report it to me, also by private email, if you wish, I will be grateful a lot!
In any case, on the first version of my paper: http://arxiv.org/vc/arxiv/papers/1105/1105.0669v1.pdf you can find an explicit proof for cases n=3 and n=5 based on my proposed technic.
What I did?
First, i heard and read that no simple prrof for Fermat's last theorem for n=3.At the same time ,i understood that correcting Euler's proof (n=3 case)is very difficult.
Secondly ,my aim was to give a simple proof for the general case.
Finally , I wanted to know how peiple look at Andrew Wiles Proof.
Taking these into account I did work on tme theorem for about three years.I think the proofe given in CMNSEM are sufficient at present,I thank you very much for the following,
(1) Anonymous can ask you queations
(2) Your commnet on CMNSEM is fare while others 'Killing' it.
(3) Above all you are a proffesor in Canada I thank you again.
For proof of FLT see http://www.coolissues.com/mathematics/Fermat/fermat.htm
Wile's proof of FLT violates the Pythagorean Theorem. see
http://www.coolissues.com/mathematics/Wile'sproofofFLT.html
For short proof of FLT see http://www.coolissues.com/mathematics/BealFermatPythagorasTriplets.html
Fermat's last theorem has been the most difficult and the most famous theorem in mathemtics.I have read the proofs of some people.All I read that unpublished in a refereed journal are incorrect.It is nonsense to argue with the authors of these proofs to my mind, since I had experienced once .Proof developed in CMNSEM is OK and based on simple mathematics.Any one can challenge against the validity of the proof.
I don't want to be too much annoying in debunking De Pedis's "proof", probably just a waste of time.
However, he tries to construct a univariate polynomial from the original trivariate statement saying "for each choice of the two other variables".
It could be legal unless, some paragraphs below, he mixes up the FLT's trivial solution with this polynomial having a single integer root.
This is clearly false because of the "for each choice of the other two variables" statement.
In any case he never use the property of the exponent "p" of being prime, and doesn't split the problem in the classic two cases (p | abc of not).
He neither takes into account the divisibility property of the binomial coefficient.
All symptoms of an hopeless attempt.
I don't know Paolo mrans Paulo Rebenboim.I have a great respect to the book of him,Fermat's Last
Theorem for Anateurs.Thanks a lot for your comment.
Answer to Paolo Tassotti post.
I don't understand the objection on first part of your post. Please could you "waste a little of your time" to explain better?
About the second part, the answers at all your objections are reported (as said in footnote 2) in the Rimbenboin book.
If you have found any error in my proof posted on arXiv.org please report it to me in clear statments, also by private email, if you wish, I will be grateful a lot!
"All symptoms of an hopeless attempt" are not proof of fault !!
Best regards
'A Simple and Short Analytical Proof of Fermar's Last Theorem' in CMNSEM is actually simple and correct and it must be re-written so that a high-scool student can understand.
I'm again asking to look at concise (1 page) description of 4 steps of offered proof of FLT. Any basic flaw will be seen from it immediately. Its URL
http://www.fileden.com/files/2012/5/4/3300464/Abstract%20of%20short%20Proof%20PDF%281%29.pdf
Regarding the South Asian Journal
of Mathematics, a number of mathematicians, including A. Razborov,
A. Abebe and N. Billor were
listed as editors by the journal
without their knowledge or consent. (Their names have now been removed).
U have received an e-mail from a man sayung that CMNSEM is not a legitmate journal.What one means by this?
The extreme breadth of the journal's coverage; the small editorial board with tiny geographical coverage and unimpressive reputations; the fact that reviews are done in two weeks; the use of Microsoft Word as the text preparation software; -- all these are red flags that something is rather odd about this journal.
I have published all of my papers on Fermat's last theorem in CMNSEM. I actually did not know any bad thing about CMNSEM.
In n=5 (with one of my students) we have made a mistake.Apart from that any one can challenge the validity of the proofs.
As far as I know CNMSEM is better tha the journal 'Advaces in pure mathematics' since ut has not published noncence.
We have corrected the case n=5 and now we have two proofs of Fermat's last theorem (FLT)for n=5 using the Identity of Fermat equation, x+y-z.
A fox, i have no any suitable word to call this man, has sent an E-mail to the authorities of my University saying that CNMSEM is not legtmate. Would you please tell us the open access journal that you edit.We are looking for the second simple proof of FLT similar to n=7 case in CMNSEM with one of my colleagues.R.A.D.P
Even CNMSEM saya 2 weeks review, they take moure than one month.
It is not nice to make fun of the intelectual capacities of someone. And he is not a system engineer, because the math he presented is on a post-graduate level , even if it has mistakes.
It is not clear who has been aimed at by the saying of anonymous of October 28.If it is Ikorong Anouk Gilbert Nemron,many will be against you because his work on Fermat's last theorem is so hilarious.
I must indorm everyone who is interested in Fermat's last theorem that,mainly, the Dean of the university of Kelaniya has decided that CNMSEM is not a refereed journal.I know that the proof of FLT in CNMSEM is correct.However, anyones comment is appreciated.
It is the Dean of the science faculty, professor in mathematics,(I dont know what he knows) as far as I know that
wanted to nullify all the papers related to FLT publisshed in CMNSEM by a lecturer of the department of mathematics of the University of Kelaniya.This is nasty and disgusting practice of this man.Let us see what will happen to the journal and proofs of FLT for many exponents.
CMNSEM is a refereed journal.
I Can prove.
Fermat,s last theorem(FLT) is the most notorious,difficult and famous theorem.If someone make(in finding a simple proof) a mistake in proving it there are a lot to laugh at it.Do these people appreciate the good work of it?.No.This is the truth.
The so called experts look at the marvelous proofs in CNMSEM with closed eyes. I an sure , the other (are being done ar present by the )proofs by the same author will make FLT very easy for all.
Harvey Friedman grand conjecture says simple mathemtics can
prove Fermat's last theorem. Also ,Colin Mactarty (philosopher)has reached the same conclusion.
One has proved Fermat's last theorem for all exponents using elementary mathematics and published in CMNSEM.Anyone can challenge the world against the validity of the proofs on behalf of the author of the papers.
I try my best to find (one is already OK)three simple proofs of Fermat's last theorem.
R.A.D.
Look at
http://www.fileden.com/files/2012/5/4/3300464/Abstract%20of%20short%20Proof%20PDF%281%29.pdf
To prove Fermat's last theorem one needs Fundamental theorem of Arithmetic and Remainder theorem (A special form) only.R.A.D.P.
Structure of Fermat triples in Mcpgr proof and proof in CMNSEM Vol.1,No.2,March 2011,pp57-63 are ninety percent coincide.One set in step.1 is wrong in Mcpgr ,therefore.I am reading the full proof however.
It is utter waste of time. I don't understand Mcpgr proof.
Take a look at:
http://www.mymathforum.com/viewtopic.php?f=40&t=40857
By now,I have read four proofs of Fermat's last theorem.Unfortunately, all are wrong.I have given up reading such proofs for sometime.
"A simple and short analytical proof of Fermat;s last theorem 'is available now.CMNSEM,Vol.2,No3. March 2011,p.57-63.
Using the fundamental theorem of Arithmetic the theorem can be proved justifying Harvey Friedman conjecture and Colin McLarty prediction.However, by no means this underestimate Andrew Wiles work,to my mind.
I thank you very much for giving me a very good chance to tell the above to the world.
R.A.D.P
Investigate one paper, and you find a network of publishers and authors working together, http://www.journalshub.com/all-journal.php, example: American Journal of Math .. http://www.journalshub.com/journal-detail.php?journals_id=127
then google for Diploma Mills, Vanity Press, and cross reference the references.
Thank you very much for your information,Anonymous.R.A.D.P.
I predicted "the death of the university" after being failed on a CELTA course in 2008.
A few weeks ago, I noticed that the rentacoder site was offering work for an assignment to be written. I e-mailed the lecturer to say that I did not want to make trouble for the person using rentacoder, but that this signalled to me the decline of civilisation as we know it.
I am very puzzled by blogs I have seen today on Fermat's Last Theorem.
There is Orwellian mention of an "open access journal". Does this mean that the journal is free to access on internet? I see now evidence that such a journal would be of poor quality.
50 years ago, we were told "publish or perish". We knew that 90% of the papers being published were rubbish.
Perhaps this is a blessing, as there is less to read. So it may be even more of a blessing if it turns out that at 99% of what is published today is rubbish.
Reading a blog on Fermats's Last Theorem. I can't get from one line to the next of Wiles's paper.But these blogs have at least the advantage of being easy to read. Maybe Wiles could republish his work in a way that is accessible to someone with only high school maths.
I will be very happy if Andrew Wiles publish
or republish his work in such a way that a High School Student can understand the proof.Then I can study something else.
R.A.D.Piyadasa
I will be very happy if Andrew Wiles publish
or republish his work in such a way that a High School Student can understand the proof.
You're going to be waiting a long time.
Thank you very much Prof Jeffrey.
I take your word.
Wish you a very happy New year as well.
R.A.D.Piyadasa
Second elementary proof of Fermat's last theorem is OK.Only high school mathematics is used.I am going to publish it.
R.A.D.Piyadasa
Here is one of some simple proofs of Fermat's last theorem as follows:
1.
X^p + Y^p ?= Z^p (X,Y,Z are integers, p: any prime >2) (1)
2. Let‘s divide (1) by (Z-X)^p, we shall get: (X/Z-X)^p +( Y/Z-X)^p ?= (Z/Z-X)^p (2)
3. That means we shall have:
X’^p + Y’^p ?= Z’^p and Z’ = X’+1 , with X’ =(X/Z-X), Y’ =(Y/Z-X), Z’ =(Z/Z-X) (3)
4. From (3), we shall have these equivalent forms (4) and (5): Y’^p ?= pX’^(p-1) + …+pX’ +1 (4)
Y’^p ?= p(-Z’)^(p-1) + …+p(-Z’) +1 (5)
5. Similarly, let’s divide (1) by (Z-Y)^p, we shall get: (X/Z-Y)^p +(Y/Z-Y)^p ?= (Z/Z-Y)^p (6)
That means we shall have these equivalent forms (7), (8) and (9):
X”^p + Y”^p ?= Z”^p and Z” = Y”+1 ,with X” =(X/Z-Y), Y” =(Y/Z-Y), Z” =(Z/Z-Y) (7)
From (7), we shall have:
X”^p ?= pY’’^(p-1) + …+pY’’ +1 (8)
X”^p ?= p(-Z”)^(p-1) + …+p(-Z”) +1 (9)
Since p is a prime that is greater than 2, p is an odd number. Then, in (4), for any X’ we should have only one Y’ (that corresponds with X’) as a solution of (1), (3), (4), (5), if X’ could generate any solution of Fermat’s last theorem in (4).
By the equivalence between X’^p + Y’^p ?= Z’^p (3) and X”^p + Y”^p ?= Z”^p (7), we can deduce a result, that for any X” in (8), we should have only one Y” (that corresponds with X’’ ) as a solution of (1),(7),(8),(9), if X” could generate any solution of Fermat’s last theorem.
X” cannot generate any solution of Fermat’s last theorem, because we have illogical mathematical deductions, for examples, as follows:
i) In (8), (9), if an X”1 could generate any solution of Fermat’s last theorem, there had to be at least two values Y”1 and Y”2 or at most (p-1) values Y”1, Y”2,…, Y”(p-1),
that were solutions generated by X”, of Fermat’s last theorem. (Please note the even number (p-1) of pY”^(p-1) in (8)). But we already have a condition stated above, that for any X” we should have only one Y” (that corresponds with X”) as a solution of (1),(7),(8),(9), if X” could generate any solution of Fermat’s last theorem.
Fermat’s last theorem is simply proved!
ii) With X”^p + Y”^p ?= Z”^p, if an X”1 could generate any solution of Fermat’s last theorem, there had to be correspondingly one Y” and one Z” that were solutions generated by X”, of Fermat’s last theorem. But let’s look at (8) and (9), we must have Y” = -Z”. This is impossible by further logical reasoning such as, for example:
We should have : X”^p + Y”^p ?= Z”^p , then X”^p ?= 2Z”^p or (X”/Z”)^p ?= 2. The equal sign, in (X”/Z”)^p ?= 2, is impossible.
Fermat’s last theorem is simply again proved, with the connection to the concept of
(X”/Z”)^p ?= 2. Is it interesting?
Sincerely yours,
Pham Duc Sinh.
Is it interesting?
No.
Thank you for letting me post the proof.
I suddenly thought of Fermat's last theorem at a night without sleep, while studying Quantum Physics and Maxwell equations years ago. And I believe and am rather sure that Fermat's last theorem was not just a hard-to-solve problem like many of us had thought of.
In fact, Fermat's last theorem is a base for setting up calculation methods for Quantum Physics and Wave equations if we merge the concept of continuity limit of real irrational numbers into rational numbers.
Please just correlate the notion of integers, rational numbers (the quantum is one, or some rational number!)to the basic concept of Quantum Physics (quantum-quanta).
And the concept of wave tranmission to the concept of continuity of real-irrational numbers!
If we apply the reasonings in the above proof of Fermat's last theorem to real-irrational numbers, we of course shall find out that the current concept of real-irrational numbers is still to be re-examined!
That is interesting, I think, because the concept of real-irrational numbers is basic for number theory, geometry, differential equations,trigonometry,...
Sinh
Sinh,
It appears to me as if your "proof" never uses the fact that X, Y, and Z are integers! So there is clearly a problem with it.
Posting it in ASCII makes it difficult to decipher. You also are missing some parenthesis, you write (X/Z-X) when you mean (X/(Z-X)). I am sorry to have wasted my time on it.
Hi George,
1.I do not know why you pose the question that I did not start with integers!
You can see that there are definition that X,Y,Z are integers, on the start of the proof! Then we have equivalent,to-be-solved equations with rational numbers.
Then is it OK to reason the proof in rational numbers, whatever they are different to integers!
We can also have reasonings for integers. But it is a little bit long to be explained. Then I avoid making it.
2. Thank you for pointing out that I did mistakes in writing
(Z/Z-X).It has to be (Z/(Z-X)). I am sorry.
I have bad eyesight and tend to abbreviate the formulas for convenience. I often make mistakes in typing!
But it does not change the thought line, does it?
Do you never make mistakes in writing mathematic formulas? I believe that people usually make.
Hi George,
Here is the typing correction.
One of some simple proofs of Fermat's last theorem as follows:
1.
X^p + Y^p ?= Z^p (X,Y,Z are integers, p: any prime >2) (1)
2. Let‘s divide (1) by (Z-X)^p, we shall get:
(X/(Z-X))^p +(Y/(Z-X))^p ?= (Z/(Z-X))^p (2)
3. That means we shall have:
X’^p + Y’^p ?= Z’^p and Z’ = X’+1 ,with X’ =X/(Z-X), Y’ =Y/(Z-X), Z’ = Z/(Z-X) (3)
4. From (3), we shall have these equivalent forms (4) and (5): Y’^p ?= pX’^(p-1) + …+pX’ +1 (4)
Y’^p ?= p(-Z’)^(p-1) + …+p(-Z’) +1 (5)
5. Similarly, let’s divide (1) by (Z-Y)^p, we shall get:
(X/(Z-Y))^p +(Y/(Z-Y))^p ?= (Z/(Z-Y))^p (6)
That means we shall have these equivalent forms (7), (8) and (9):
X”^p + Y”^p ?= Z”^p and Z” = Y”+1 ,with X” =X/(Z-Y), Y” =Y/(Z-Y), Z” = Z/(Z-Y) (7)
From (7), we shall have:
X”^p ?= pY’’^(p-1) + …+pY’’ +1 (8)
X”^p ?= p(-Z”)^(p-1) + …+p(-Z”) +1 (9)
Since p is a prime that is greater than 2, p is an odd number. Then, in (4), for any X’ we should have only one Y’ (that corresponds with X’) as a solution of (1), (3), (4), (5), if X’ could generate any solution of Fermat’s last theorem in (4).
By the equivalence between X’^p + Y’^p ?= Z’^p (3) and X”^p + Y”^p ?= Z”^p (7), we can deduce a result, that for any X” in (8), we should have only one Y” (that corresponds with X’’ ) as a solution of (1),(7),(8),(9), if X” could generate any solution of Fermat’s last theorem.
X” cannot generate any solution of Fermat’s last theorem, because we have illogical mathematical deductions, for examples, as follows:
i) In (8), (9), if an X”1 could generate any solution of Fermat’s last theorem, there had to be at least two values Y”1 and Y”2 or at most (p-1) values Y”1, Y”2,…, Y”(p-1),
that were solutions generated by X”, of Fermat’s last theorem. (Please note the even number (p-1) of pY”^(p-1) in (8)). But we already have a condition stated above, that for any X” we should have only one Y” (that corresponds with X”) as a solution of (1),(7),(8),(9), if X” could generate any solution of Fermat’s last theorem.
Fermat’s last theorem is simply proved!
ii) With X”^p + Y”^p ?= Z”^p, if an X”1 could generate any solution of Fermat’s last theorem, there had to be correspondingly one Y” and one Z” that were solutions generated by X”, of Fermat’s last theorem. But let’s look at (8) and (9), we must have Y” = -Z”. This is impossible by further logical reasoning such as, for example:
We should have : X”^p + Y”^p ?= Z”^p , then X”^p ?= 2Z”^p or (X”/Z”)^p ?= 2. The equal sign, in (X”/Z”)^p ?= 2, is impossible.
Fermat’s last theorem is simply again proved, with the connection to the concept of
(X”/Z”)^p ?= 2.
Sinh,
In your "proof", nowhere do you need to use the fact that X,Y and Z are integers. Therefore, your proof also shows that X^p + Y^p = Z^p has no solutions when p>2 is prime and X, Y and Z are arbitrary real numbers! This is certainly false, therefore, your "proof" must be incorrect.
George,
It is very interesting to discuss this. I wrote to Jeffrey Shallit that we can also apply this proof to real-irrational numbers, and find out abnormal new results for real-irrational number.
When the proof can be applied to whatever type of number, it is correct with that type of number! The main things to be checked are in the logical reasonings and the variables, too.
Please read:
ii) With X”^p + Y”^p ?= Z”^p, if an X”1 could generate any solution of Fermat’s last theorem, there had to be correspondingly one Y” and one Z” that were solutions generated by X”, of Fermat’s last theorem. But let’s look at (8) and (9), we must have Y” = -Z”. This is impossible by further logical reasoning such as, for example:
We should have : X”^p + Y”^p ?= Z”^p , then X”^p ?= 2Z”^p or (X”/Z”)^p ?= 2. The equal sign, in (X”/Z”)^p ?= 2, is impossible.
With your question, we can rewrite that:
With X" and Z" are rational numbers (they come from the definition that X,Y,Z are integers), the equal sign in (X”/Z”)^p ?= 2 is impossible.
(The problem (X”/Z”)^p ?= 2 has been one of the basic excercises to be proved in number theory and to be examined for the concept of real-irrational numbers).
I am happy and delighted that I can disscuss Fermat's last theorem with you.
WE WOULD HAVE A LOT TO DISCUSS IN NUMBER THEORY, GEOMETRY, DIFFERENTIAL AND INTEGRAL CALCULUS,..., DIFFERENTIAL EQUATIONS,..., ALGEBRA,...,TRIGONOMETRY,..., QUANTUM PHYSICS,...ETC
I CAN ASSURE YOU THAT!
(And I still avoid working with complex-imaginery numbers. I have different thoughts in definitions of complex-imaginery numbers!)
And this is another version of the proof:
Fermat's last theorem has an equivalent theorem (we accept the case X^4 +Y^4 ?= Z^4. It has been proved):
X^p+Y^p ?= Z^p (X,Y,Z are rational numbers, p : any prime>2). (1)
From (1)we can use the reasonings
of the above to prove that (1) is equivalent to the following theorem:
X’^p + Y’^p ?= Z’^p and Z’ = X’+1
X',Y',Z' are rational numbers.
That means:
Y’^p ?= pX’^(p-1) + …+pX’ +1 (3).
We need checking (2) and (3) only with Y'(integers) and X' (real rational numbers that cannot be reduced to integers), then we can see that the equal sign in (3) is impossible.
George,
If you are still in doubt that the current concepts of real irrational numbers are still to be re-examined, let me post an exercise of real numbers that I have read in a number textbook as follows:
Let a,b,c,d are rational numbers, e is real irrational number, prove that if a+ce= b+de, then we must have a=b and c=d.
Proof:
(a-b) ?= (d-c)e. Then we can easily see that a=b and c=d, otherwise e ?= (a-b)/(d-c). An irrational number cannot be equal to a rational number by definition.
We can see that, even with basic calculations such as addition and multiplication, irrational numbers behave not quite the same as rational numbers . This exercise reminds us the independence concept of vectors in linear algebra.
Then, the posted proof of Fermat's last theorem shows us that there is actually a new hidden difference between rational and irrational numbers.
If we want to deny a proof, we must point out the false and illogical reasoning or false, illogical variables used in the proof.
We cannot arbitrarily apply a proof or function used for rational numbers to irrational numbers, right?
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