## Fermat's Last Theorem

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• To Professor Chris Caldwell and members of the Primenumbers Yahoo Group. Hello to all, I am very happy to present my proof of Fermat s Last Theorem
Message 1 of 19 , Aug 30, 2001
To Professor Chris Caldwell and members of the Primenumbers Yahoo
Group.

Hello to all,
I am very happy to present my proof of Fermat's Last Theorem
http://members.tripod.co.uk/comms1/fidn1/fermat.html

This proof has been submitted to the Royal Society of London,
England. I would appreciate any comments, typos or corrections.
Thank you. The nature of the lengthy, but straightforward proof, is
that there may be `slight' logic slips, similar to the `bunker' that
Andrew Wiles found with his tee shot. But as with his proof I don't
anticipate too many problems fixing it, as the proof is very
structured and uses classical techniques. So, get yourself a cold
beer, sit back and enjoy the fun!

Regards,
Paul Mills,
England.
• ... In lemma 3.2 you say otherwise x, y, z would not be pairwise coprime , however nowhere have you stated that they ought to be pairwise coprime (note that
Message 2 of 19 , Aug 30, 2001
On Thu, 30 August 2001, paulmillscv@... wrote:
> I am very happy to present my proof of Fermat's Last Theorem
> http://members.tripod.co.uk/comms1/fidn1/fermat.html

In lemma 3.2 you say "otherwise x, y, z would not be pairwise coprime", however nowhere have you stated that they ought to be pairwise coprime (note that it is the first mention of the term in the document).

Then you procede to perform "division".

By this stage are you in a ring or a field?
Either way, which one?

To be honest, this kind of post is better off discussed on the newsgroup sci.math, where there are several people who debunk two elementary proofs of FLT before breakfast each day.

Phil
Phil

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• In Lemma 3.2, section (1), you calculate d=z-y, then proceed to divide y by d and set this value to t. From then on you presume t to be an integer, but it is
Message 3 of 19 , Aug 30, 2001
In Lemma 3.2, section (1), you calculate d=z-y, then proceed to divide y by
d and set this value to t. From then on you presume t to be an integer, but
it is not necessarily one.

Tom

-----Original Message-----
From: paulmillscv@... [mailto:paulmillscv@...]

Hello to all,
I am very happy to present my proof of Fermat's Last Theorem
http://members.tripod.co.uk/comms1/fidn1/fermat.html
• Phil Carmody wrote ... Oh dear, Phil, I think you may just have contradicted one of the top 10 number theorists in the world:
Message 4 of 19 , Aug 30, 2001
Phil Carmody wrote

> To be honest, this kind of post is better off discussed on the
> newsgroup sci.math, where there are several people who debunk
> two elementary proofs of FLT before breakfast each day.

Oh dear, Phil, I think you may just have contradicted
one of the top 10 number theorists in the world:

http://members.tripod.co.uk/comms1/fidn1/world1.html

> Their deliberations in number theory matters is considered
> authoritative according to the best of their present
> knowledge and belief.

Who are the likes of thee and me to try to educate
one of this "Curia" ?

David
• I guess you are wrong Phil. Just below (1.1) it says that the x,y,z are assumed to be pairwise relatively prime Hugo Scolnik If we knew what it was we were
Message 5 of 19 , Aug 30, 2001
I guess you are wrong Phil. Just below (1.1) it says that the x,y,z are
assumed to be pairwise relatively prime

Hugo Scolnik

"If we knew what it was we were doing, it would not be called research,
would it?"
• Dear all, In fact, I saw the same major problem mentioned below when I was reading Paul Mills proof of Fermet s Last Theorem this morning. Also, in Lemma
Message 6 of 19 , Aug 30, 2001
Dear all,

In fact, I saw the same major problem mentioned below when I was reading
Paul Mills' "proof" of Fermet's Last Theorem this morning. Also, in Lemma
3.2 on page 3, "Dividing equation 3.6 by d" should read "Dividing 3.6 by
d^p". This is just a minor typo.

At present moment, I don't see how you (Mills) can easily fix the problem
mentioned below (t is not necessarily in Z_p).

Thanks!

Regards,
T. W. Alan Wong

On Thu, 30 Aug 2001, Hadley, Thomas H (Tom), NLCIO wrote:

> In Lemma 3.2, section (1), you calculate d=z-y, then proceed to divide y by
> d and set this value to t. From then on you presume t to be an integer, but
> it is not necessarily one.
>
> Tom
>
> -----Original Message-----
> From: paulmillscv@... [mailto:paulmillscv@...]
>
> Hello to all,
> I am very happy to present my proof of Fermat's Last Theorem
> http://members.tripod.co.uk/comms1/fidn1/fermat.html
>
>
>
> Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
> The Prime Pages : http://www.primepages.org
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>
• ... Thanks Hugo. Bells started ringing on the lines below, so I rewound a few lines to make sure I had the full context, and for some strange reason the word
Message 7 of 19 , Aug 30, 2001
On Thu, 30 August 2001, "Hugo Scolnik" wrote:
> I guess you are wrong Phil. Just below (1.1) it says that the x,y,z are
> assumed to be pairwise relatively prime

Thanks Hugo. Bells started ringing on the lines below, so I rewound a few lines to make sure I had the full context, and for some strange reason the word coprime jumped out at me, I thought I'd not seen it before, did a text-search of the document for both 'coprime' and 'co-prime', and came to the conclusion that I was right in not seeing the concept introduced before. I was wrong. I think that I was skipping over the first few sections too quickly as it was stuff that I had seen before. If only I'd searched for 'pairwise', I'd have found it.

Phil

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• Dear friends ... Let us try to gain something from the experience. Find a counterexample to Mr Mills statement that (x+y)^p = x^p + y^p + pX = X is an
Message 8 of 19 , Aug 30, 2001
Dear friends

Since Paul Mills wrote:

>Hello to all,
> I am very happy to present my proof of Fermat's Last Theorem
> http://members.tripod.co.uk/comms1/fidn1/fermat.html

Let us try to gain something from the experience. Find a counterexample
to Mr Mills' statement that

(x+y)^p = x^p + y^p + pX => X is an integer, not divisible by p.

Since Dr Mills recently posed a problem to France and imposed a time
limit, I would like to offer Dr Mills a similar time limit to find the
error.

Readers of these pages not included in the number theory world rankings
may use the clue at

http://www.utm.edu/research/primes/glossary/WieferichPrime.html

instead, and their time limit is three seconds.

Chris Nash
Lexington KY
UNITED STATES
• Dear Chris, Hello. I apologise, I knew I had missed someone from the rankings, Chris Mr Primeform Nash. Your comment is interesting and points to a `main
Message 9 of 19 , Aug 31, 2001
Dear Chris,
Hello. I apologise, I knew I had missed someone from the
rankings, Chris 'Mr Primeform' Nash. Your comment is interesting and
points to a `main point' of the proof straightaway.
By Lemma 3.2 equations 3.6 and 3.7 have no solutions.
Therefore, with z = aq, (z,p) =1, x and y are also co-prime to
p. Therefore any term x^(p-j)y^j in the Binomial expansion of (x
+ y)^p has no factor of p and the integer X is not divisible by p.
Then we go into equation 3.7 and deduce a contradiction.

Regards,
Paul Mills
England.
• ... Counterexample (Wieferich, 1909): x=1, y=1, p=1093. You are 90 years out of date; please retract. David
Message 10 of 19 , Aug 31, 2001
Paul: Chris Nash has clearly identified an error:
> (x+y)^p = x^p + y^p + p*X (3.37)
> Where X is an integer, not divisble by p.
Counterexample (Wieferich, 1909): x=1, y=1, p=1093.
You are 90 years out of date; please retract.
David
• Hello to all, Thank you for your critical responses to my posting of Fermat s Last Theorem. I have gathered the results together in a notes paper at this
Message 11 of 19 , Sep 1, 2001
Hello to all,
Thank you for your critical responses to my posting of Fermat's
Last Theorem. I have gathered the results together in a notes paper

http://members.tripod.co.uk/comms1/papers/fltnotes.pdf

Could you please indicate the step (A-G) you are on when you make a
critical comment so I know where to place it and the reply in this
notes paper. All notes updates will be placed in this paper. Again,

Regards,
Paul Mills
England.
• Hi folks, The administrative assistants of many math faculties throughout the world are often deluged with proofs of FLT/Goldbach/Riemann/whatever. The only
Message 12 of 19 , Sep 1, 2001
Hi folks,

The administrative assistants of many math faculties throughout the
world are often deluged with "proofs" of FLT/Goldbach/Riemann/whatever.
The only way they can suitably handle the workload is a stock letter as
follows.

Dear __________,

Thank you for your generous submission of your proposed proof of ____.
An error has been found on page __ line __. Please correct and
resubmit.

Should anyone pass through the AA's office (professors, alumni,
undergraduates, tramps) they could cast a cursory glance over the
paper, take a letter, fill in the blanks. The letters go into a file,
and should the candidate resubmit, there was usually another letter
left in the file ready to send. Eventually the war of attrition ends by
the candidate giving up on the proof attempt. In this way neither the
faculty members nor the AA needed to do excessive or unwarranted work.
It's very rare for anything to get through this process and require
significant attention.

It wouldn't take anyone with a little Web savvy much work at all to
write a script to generate such letters (perhaps indeed one already
exists). Unless this debacle is resolved quickly, I may do so.

In other words, unless candidates desire to receive e-mail continually
from a web bot, they'd better start listening to objections that are
generously offered in good faith. Many members of the list have quite
high tolerance for this sort of thing. Others, however, have little or
zero.

Chris
• On Sat, 01 September 2001, Chris Nash wrote: [SNIP - sage words on FLT proofs ] Personally, I view FLT as a) proved already. b) _off-topic_ for a _primes_
Message 13 of 19 , Sep 1, 2001
On Sat, 01 September 2001, "Chris Nash" wrote:
[SNIP - sage words on FLT "proofs"]

Personally, I view FLT as
b) _off-topic_ for a _primes_ list.

I say the latter as I participate in 3 other more general mathematical online groups on which FLT discussions are far more on-topic. To say that FLT is related to primes as one only has to prove prime cases (with 2^2 of course) is to say that Douglas Adam's 42 is related to primes as it's 2.3.7. I don't deny that the history of FLT has spawned several interesting prime-related concepts (such as Sophie Germain primes), but those spawnings were a hundreds of years ago now, and the concepts have their own independent life now, independent of FLT.

Usenet's sci.math is _the_ place for amateur FLT proofs.

Writing solely as an ordinary list member, nothing more.

Phil

Mathematics should not have to involve martyrdom;
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• Phil Carmody wrote ... Of which Wieferich s is one. Paul Mills may be excused for being 90 years off the pace, but his reluctance to admit that Wieferich kills
Message 14 of 19 , Sep 1, 2001
Phil Carmody wrote

> the history of FLT has spawned several interesting
> prime-related concepts

Of which Wieferich's is one.

Paul Mills may be excused for being 90 years off
the pace, but his reluctance to admit that
Wieferich kills his paper stone dead is lamentable.

? x=3860385325121537;
? y=4047907402674639732737;
? p=3511;
? X=((x+y)^p-x^p-y^p)/p;
? if(X%p==0,print("Paul Mills' paper is plain wrong"));
Paul Mills' paper is plain wrong
• Hello to all, Here is version 2 of my proof of Fermat s Last theorem. Am I on the green or a greenside bunker? Have fun.
Message 15 of 19 , Sep 6, 2001
Hello to all,
Here is version 2 of my proof of Fermat's Last theorem. Am I on
the green or a greenside bunker? Have fun.

http://members.tripod.co.uk/comms1/papers/flt2.pdf

Of course, FLT is relevant to primes because if you can prove x^p +
y^p = z^p for odd primes p then FLT is proved.

Regards,
Paul Mills,
England.
• ... Couldn t the same be said for the numbers q(n), where q(n) is the number of cards in Nathan Russell s nth Magic the Gathering deck? Nathan
Message 16 of 19 , Sep 6, 2001
On Thu, 06 Sep 2001 11:46:11 -0000, paulmillscv@... wrote:

>
>Hello to all,
> Here is version 2 of my proof of Fermat's Last theorem. Am I on
>the green or a greenside bunker? Have fun.
>
>http://members.tripod.co.uk/comms1/papers/flt2.pdf
>
>Of course, FLT is relevant to primes because if you can prove x^p +
>y^p = z^p for odd primes p then FLT is proved.

Couldn't the same be said for the numbers q(n), where q(n) is the
number of cards in Nathan Russell's nth Magic the Gathering deck?

Nathan
• ... Don t confuse the tool and the application . FLT proofs may involve primes, but that doesn t mean that FLT is _about_ primes. FLT is about _all positive
Message 17 of 19 , Sep 6, 2001
On Thu, 06 September 2001, paulmillscv@... wrote:
> Hello to all,
> Here is version 2 of my proof of Fermat's Last theorem. Am I on
> the green or a greenside bunker? Have fun.
>
> http://members.tripod.co.uk/comms1/papers/flt2.pdf
>
> Of course, FLT is relevant to primes because if you can prove x^p +
> y^p = z^p for odd primes p then FLT is proved.

Don't confuse the 'tool' and the 'application'.
FLT proofs may involve primes, but that doesn't mean that FLT is _about_ primes. FLT is about _all positive integers_.
One could say that _every_ theorem that _anywhere_ uses properties of UFDs is 'prime related'.

Apart from that, it appears that all you've done is add some handwaving to 3.2, and created a bizarre mathematical structure, the �modulo rational plane�, which you do not prove to be either a Ring or Field. You do not even claim it to be either of those, and the operations you perform in the structure I consider to be inconsistent with both.
The few "definitions" that you do give are either inconsistent with the usage of the things you are trying to define, or are so ambiguously worded that they don't actually define anything.
e.g. "Note that modulo rationals are defined in pairs." causes nothing but confusion at this end.

Can you _please_ move this to sci.math, In particular you appear to have reached a similar position to where the resident FLT prover got to about 2 weeks ago, before he flew off at a normal (not a tangent), so all the relevant arguments are fresh in the debunkers' minds.

It appears a de-'bunker' is what is required...

Phil

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• Recently I completed a paper to prove Fermat s Last Theorem. I would like a feed back from members of this group before trying to publish the paper. Thank you.
Message 18 of 19 , Oct 20, 2013
Recently I completed a paper to prove Fermat's Last Theorem. I would like a feed back from members of this group before trying to publish the paper.

Thank you.

{eter/

[Non-text portions of this message have been removed]
• I will have to put the paper on the Interne; and then send another message. Peter ... From: Peter Lesala To: primenumbers@yahoogroups.com Sent: Sunday, October
Message 19 of 19 , Oct 20, 2013
I will have to put the paper on the Interne; and then send another message.

Peter
----- Original Message -----
Sent: Sunday, October 20, 2013 8:27 PM