sci.math FAQ: Status of FLT
Alex Lopez-Ortiz
Last-modified: December 8, 1994
Version: 6.2
What is the current status of FLT?
Andrew Wiles, a researcher at Princeton, claims to have found a proof.
The proof was presented in Cambridge, UK during a three day seminar to
an audience which included some of the leading experts in the field.
The proof was found to be wanting. In summer 1994, Prof. Wiles
acknowledged that a gap existed. On October 25th, 1994, Prof. Andrew
Wiles released two preprints, Modular elliptic curves and Fermat's
Last Theorem, by Andrew Wiles, and Ring theoretic properties of
certain Hecke algebras, by Richard Taylor and Andrew Wiles.
The first one (long) announces a proof of, among other things,
Fermat's Last Theorem, relying on the second one (short) for one
crucial step.
The argument described by Wiles in his Cambridge lectures had a
serious gap, namely the construction of an Euler system. After trying
unsuccessfully to repair that construction, Wiles went back to a
different approach he had tried earlier but abandoned in favor of the
Euler system idea. He was able to complete his proof, under the
hypothesis that certain Hecke algebras are local complete
intersections. This and the rest of the ideas described in Wiles'
Cambridge lectures are written up in the first manuscript. Jointly,
Taylor and Wiles establish the necessary property of the Hecke
algebras in the second paper.
The new approach turns out to be significantly simpler and shorter
than the original one, because of the removal of the Euler system. (In
fact, after seeing these manuscripts Faltings has apparently come up
with a further significant simplification of that part of the
argument.)
The preprints were submitted to The Annals of Mathematics. According
to the New York Times the new proof has been vetted by four
researchers already, who have found no mistake.
In summary:
Both manuscripts have been accepted for publication, according to
Taylor. Hundreds of people have a preprint. Faltings has simplified
the argument already. Diamond has generalised it. People can read it.
The immensely complicated geometry has mostly been replaced by simpler
algebra. The proof is now generally accepted. There was a gap in this
second proof as well, but it has been filled since October.
You may also peruse the AMS site on Fermat's Last Theorem at:
gopher://e-math.ams.org/11/lists/fermat
_________________________________________________________________
alop...@barrow.uwaterloo.ca
Tue Apr 04 17:26:57 EDT 1995
Archimedes Plutonium
alop...@neumann.uwaterloo.ca (Alex Lopez-Ortiz) writes:
>
> What is the current status of FLT?
Well, I am glad you asked Alex.
FLT and a mountain of Number theory conjectures were backlogged. In
fact the two oldest problems in all of math are two number theory
problems concerning perfect numbers. Now, a reasonable person, whether
in math or and outsider to math would ask the obvious question why the
backlog in Number theory? For it is just a branch of math. And why the
difficulty with FLT, and Goldbach conjecture and a whole pile of simple
to understand conjectures by which have escaped the greatest math
people like Euler, Gauss, Riemann?
The answer lies in the very concept of what is a number which makes
the subject of Number theory. Number theory conjectures are no more
difficult to prove than any other branch of math once the understanding
of what is a Number for Number theory. Until 1993, everyone thought a
Natural number = finite integer. For example 1, 2, 3, 4 and on and on.
That naive concept of Natural number was embellished in the Peano
axioms.
Then in 1993, with the help of a circle of smart posters, it was
found out that the concept of a finite integer is smoke and mirrors.
Reals are infinite, that is, take any Real number and it is infinite
string rightwards. Sure you can write the Real 1/3 but it is .3333.....
Every Real is infinite. Infinity was the bugaboo. The ill defined
concept of Natural number was why noone could prove Pierre Fermat's
conjecture nor ever will with Natural = Finite Integer.
The true Naturals have an infinite machinery to each and every
Natural number. That is what I mean by Naturals = Adics = Infinite
Integers. The moment you, or anyone asks for ALL the Naturals, like
what Fermat asks for in his FLT. ALL, all Naturals means Naturals that
never end. Means numbers which are infinite. If Pierre had said FLT for
Naturals ....0000001 , .....00002, ...122211122. etc his conjecture
(provided he knew adics) would have been solved long long time ago.
It was never the case that this conjecture of Pierre Fermat was
difficult. It was extremely hard though to come to the final
realization that our understanding of what a Natural number really is.
All numbers that exist, in order for the number to exist, must have the
ingredient of infinity in it. Just because of the simple fact that
every number has PLACE VALUE, and that place value is infinite in
extent.
Here is FLT proved, only surprize, it is not true as Pierre conjectured
but just the reverse. It is false for all exponents, not just 2 has
counterexamples. This revolution in math is not unlike the revolution
in physics where quantum physics replaced Newton physics. This is
reasonable, surely math people are not so arrogant as to think that
their definition and concept of number is to last absolutely as in
absolute space and time of Newton.
The expression a^n+b^n=c^n is true for all n, in 10-adics,
given the following
values.
a= ...9977392256259918212890625
b= ...0022607743740081787109376
c= ...0000000000000000000000001
Those are Naturals, just as the Natural number 231 which is ...0000231.
ABRacadabra--- all the mountains of Number theory conjectures disappear
and Number theory is no longer a bugaboo
Dan Cass
we make the identification "naturals = adics".
I propose another identification which would make
even faster mincemeat out of lots of conjectures:
NATURALS = REALS.
Then FLT becomes false, since for any n there are
nonzero reals satisfying x^n + y^n = z^n.
And if a prime number is one whose only divisors
are itself and 1 then in this naturals=reals world
there are no primes at all, so that most statements
about primes become true vacuously.
--dmc
Archimedes Plutonium
d...@sjfc.edu (Dan Cass) writes:
> And if a prime number is one whose only divisors
> are itself and 1 then in this naturals=reals world
> there are no primes at all, so that most statements
> about primes become true vacuously.
In adics there are no primes either. But statements about primes would
more likely be false vacuously.
BTW, true vacuously makes sense, but is there such a beast as false
vacuously?
Brian Conrad
>we make the identification "naturals = adics".
>I propose another identification which would make
>even faster mincemeat out of lots of conjectures:
> NATURALS = REALS.
>Then FLT becomes false, since for any n there are
>nonzero reals satisfying x^n + y^n = z^n.
>are itself and 1 then in this naturals=reals world
>there are no primes at all, so that most statements
>about primes become true vacuously.
Hey, what a great idea! That just solved my thesis problem too.
Thanks! :)
Brian
Dan Cass
>In article <1995May7.1...@sjfc.edu>
>d...@sjfc.edu (Dan Cass) writes:
>
>> And if a prime number is one whose only divisors
>> are itself and 1 then in this naturals=reals world
>> there are no primes at all, so that most statements
>> about primes become true vacuously.
>
>more likely be false vacuously.
>
>BTW, true vacuously makes sense, but is there such a beast as false
>vacuously?
If the statement is universally quantified over primes
and there ain't any, it becomes true vacuously.
But a statement which asserts the existence of primes
with certain properties becomes false; I suppose one
could say "false vacuously" for this, though I haven't
heard that phrase used. And some of the common
conjectures that come to mind, like Goldbach's
"every even integer >= 4 is the sum of two primes"
are of the latter type, and so become false.
A big chunk of FLT is covered by the following statement:
For every odd prime p the equation
x^p + y^p = z^p
has no solutions in nonzero integers x,y,z.
If there are no primes, it would seem this version
becomes true vacuously.
I guess the negation of any statement which is
"true vacuously" could be called "false vacuously".
But who can say what is happening in an empty universe?
Maybe a buddhist.
--dmc
Mark Meyer
AP> Until 1993, everyone thought a Natural number = finite integer.
And after 1993, everyone but AP thought "natural number" =
"finite integer". Not much of a revolution, really.
--
Mark Meyer | mme...@dseg.ti.com |
Texas Instruments, Inc., Plano, TX +--------------------+
Every day, Jerry Junkins is grateful that I don't speak for TI.
What if there were no hypothetical situations?
Archimedes Plutonium
mme...@skips0.dseg.ti.com (Mark Meyer) writes:
> >>>>> "AP" == Archimedes Plutonium <Archimedes...@dartmouth.edu> writes:
> AP> Until 1993, everyone thought a Natural number = finite integer.
>
> And after 1993, everyone but AP thought "natural number" =
> "finite integer". Not much of a revolution, really.
And circa 1900, there was only Tesla with AC rising up against the
forces of Edison, only Tesla, and then a champion (Westinghouse) came
in to up the total of 2 people who realized AC was superior and true.
In 1993 there was only one who knew Naturals = Adics = Infinite
Integers, (if others, they are keeping quiet), and in ____ came a
champion in to up the total of 2 and rise above the forces of apathy
and math stupidity.
Archimedes Plutonium
> And after 1993, everyone but AP thought "natural number" =
> "finite integer". Not much of a revolution, really.
Mark Meyer, Nigel Cole, and Brian Conrad who so fervently believe in
Naturals = Finite Integers just as their mentors A. Wiles, J H Conway,
Bobby Langlands, and although this second group of trios may seem
more respectable as math persons. Wiles, Conway, Langlands may strike
you as more respectable than Meyer, Cole, or Conrad but I don't think
that is a good reason to pick on Cole. And below I do not pick on him
for I broadcast as to how Cole speaks for the entire math world of his
brilliant definition of what finite number means and then he goes on to
prove that Naturals = Finite Integers, not my teachings--- Naturals =
Adics = Infinite Integers. What Nigel Cole posted in 1993 is what
Meyer, Conrad, Wiles, and Conway strongly believe in and it is the way
math has always been done up to now. John on his visit here to
Dartmouth in 1993 even asked how can I possibly add 1 to an Infinite
Integer? Good question for me in 1993 because I had never heard of
Adics yet. I cannot speak for Bobby, but I think he belongs with the
bandwagon.
Go get them Nigel. That don't rhyme. Go get them Bobby, that sounds
much better.
----start of where Nigel nails down 6 proofs which the math world
endorses as the last words on the topic of Naturals = Finite
Integers--------------
Newsgroup: sci.math
From: nhc...@zebekia.demon.co.uk (Nigel Cole)
Subject: Re: World's first proof of Fermat's Last Theorem and
References: <2q772k$n...@dartvax.dartmouth.edu>
Organization:
Lines: 176
Date: Mon, 9 May 1994 22:06:04 +0000
Message-ID: <768548...@zebekia.demon.co.uk>
OK, I'll let myself be sucked into this entertaining morass :-)
In article <2q772k$n...@dartviz.dartmouth.edu>
Ludwig.P...@dartmouth.edu (Ludwig Plutonium) writes:
> Plutonium Axioms:
> P1) There exists two numbers 2 and 1.
> P2) 2 =1+1
> P3) From P2, having been given equality and addition, new numbers > are manufactured by adding 1.
> P4) If a set of numbers contains 2=1+1, and all the numbers
> manufactured by adding 1, then it is the set of all Whole Numbers.
>
> Definition of Whole numbers: all possible infinite digit strings >leftward of the radix point and where all strings rightward are >finite.
> Notice that P-adics and N-adics are all subsets of the Whole >numbers.
Flaw: These axioms , as given, can only generate the Natural (finite)
numbers. At each stage of P4, the new number is created by adding 1 to
a finite number. At no point can this generate an infinite number.
Illustration: At each stage, represent the number in its own base. Once
you have P2, all further numbers are represented as 10, with the base
varying. At no point do you generate a 3-digit number, let alone an
infinite number.
> (1) Consider the Naturals with their endless strings of 0's to the >left of the last digit. Observe that the Naturals are a proper subset >of the P-adics. For example the Naturals 1, 94, and 231 are P-adics
> . . .001, and . . .0094. and . . .00231. respectively.
OK so far. The Naturals are trivially a subset of the infinite numbers.
> The Naturals never have a largest member by the Math Induction >postulate. The P-adics do have a largest member as demonstrated >by . . .999. The number . . .999. is the last of all possible digit >arrangements. There exists no other maximal possibe
digit >arrangements.
So what is . . .999 + 1? By P4, it must be possible to do this
addition. If it is 0, arithmetic with the P-adics is arithmetic modulo
. . .999 + 1, so it cannot be used with FLT, which uses "normal"
arithmetic (ie not modulo any number). FLT is trivially false when used
with arithmetic modulo some number (eg. a^b + a^b \equiv a^b modulo
a^b).
> The Naturals are Whole Numbers and the P-adics are Whole >numbers. Since the P-adics have a largest member and the Naturals >do not then the P-adics are a proper subset of the Naturals.
This implies that there exists a Natural number which is not a P-adic,
which is a direct contradiction to your definition that all Natural
numbers are P-adics.
The flaw is the hidden assumption that the largest member of the
P-adics is a Natural number - it is, of course, trivial to "prove" a
fact once you have assumed it to be true! If the largest number is not
a Natural number, then all you can prove is that the finite numbers are
a subset of the infinite ones.
> ((2)) Consider factorials. Notice that as you take the factorial of >every consecutive Natural number starting with 1 that 5! has the >last digit 0. That 10! has the last two digits 0's, and 15! has more >last digits 0 and so on. As the numbers
become larger and larger the >factorial of these larger numbers
continues to have more 0's as the >last digits. In the limit, n! is
the number . . .000. which is 1 larger >than the largest P-adic . .
.999.
n! = nx(n-1)!, so n-1 must already be an infinite number for n! to be
one. At no stage can manipulating finite numbers in this way generate
an infinite one.
And what is . . .000? If it is a P-adic, then you have a P-adic greater
than the largest P-adic, which is a contradiction. If it is a finite
number (eg zero), then there is a largest Natural number, which is also
a contradiction.
> Let us deal with the binary system and apply it to this definition >of Whole Number. Notice that in the application of the Peano Axioms >to the definition of Whole Number in binary system that Math >Induction counts every one of those binary infinite
leftward >strings. Notice that the binary system makes the string . .
.1111. >equal to the string . . .999. of base 10.
>Notice that in binary there are no gaps of levels.
False: this ignores the gap between infinite strings and finite ones.
There is no way of deciding the smallest infinite string that is not a
finite string.
> What is the inverse of 0? Answer is infinity. If there are more >than one kind of infinity means there are more than one kind of 0. >Impossibility for all of math is then destroyed.
Wrong: the inverse (under multiplication is undefined - that is part of
the definition of a field. It is *NOT* any form of infinity. And it is
certainly not any form of Natural number.
The proof that it is not a Natural number is trivial: 0n = 0 for any
Natural number n; a hypothetical inverse X would need to obey 0x = n.
But, if X is a natural number, then 0x=0. Thus, X cannot be a natural
number. Proving in the other direction: if X was a natural number
satisfying 0X=n, then 0=n/X unless X=0. But, the Natural numbers are
all finite strings, so n/X must be greater than 0. Alternatively, if X
is a Natural number >1 satisfying 0X=n, then there exists a Natural
number X-1. 0(X-1) = 0X-0 = 0X = n. But that makes X-1 an inverse too.
Repeating the process shows that, if X was an inverse, all numbers
smaller than X would be inverses too, which contradicts the assumption
of a unique inverse.
> ((((4)))) FOURTH PROOF THAT THE PEANO AXIOMS GO INTO THE >INFINITE INTEGERS AND P-ADICS. If false then there exists two >kinds of infinities. One for Counting numbers and one for Reals. >Two kinds of infinities implies two kinds of the number 0.
False: Two kinds of infinities do *not* imply two kinds of 0. They are
not inverses of one-another.
> (((((5))))) 5th Proof that the Peano Axioms yield the P-adics or >Infinite Integers.
[Long section of geometric gibberish deleted]
By switching from a semi-infinite line in an infinite plane to a finite
line on a sphere, all you prove is that arithmetic modulo some number
is not the same as normal arithmetic. Arguing by real-world analogy is
spurious.
> and multiplication. Complete in subtraction because . . .999999999. >is -1
>and . . . . .9999999998. is -2, and so on. Complete in division when >considering Wave number mirror images.
Good - you agree that . . .999999 + 1 \equiv 0. So you are doing all
your arithmetic modulo . . .999999 +1.
> ((((((6)))))) This constitutes the 6th proof that the Peano Axioms >yield Infinite Integers.
[Non-existent proof omitted :-)]
> Related to physics, FLT states that space is quantized.
No it doesn't. It doesn't say anything about space. The quantization of
space has nothing to do with FLT - space may or may not be quantized in
complete independence of whether FLT is true or not.
PROOFS THAT INFINITE NUMBERS ARE NOT NATURAL NUMBERS
A) Consider an infinite number consisting of a repeating (finite)
sequence of n digits in base b, n>0, b>1. Call the number N, and the
sequence <n>.
N = ...<n><n><n>. N is a Natural number, so N>0.
b^n N = ...<n><n><n zeros>
b^n N + <n> = ...<n><n><n> = N
So (b^n - 1)N = -<n>
So (b^n - 1)N < 0
So N<0. Contradiction. So N cannot be a Natural number (or, indeed,
obey the laws of arithmetic).
B) The process of generating the Natural numbers imposes an ordering.
Hence, given two different natural numbers A and B, it is always
possible to determine whether A<B or B<A - the number which was
produced first by the generating process is the smaller number. This is
not true of the infinite numbers:
Let A = . . .01010101, B = . . .10101010
10A = . . .10101010 = B, so A<B
10B+1 = . . .1010100 + 1 = . . .1010101 = A, so B<A
So A<B and B<A, contradiction. So normal arithmetic operations and
relations are not valid for the infinite numbers. So they are not
Natural numbers.
C) No 2 different Natural numbers are equal to one another, since they
are all at least 1 away from their nearest neighbours. This property
holds whatever number base is chosen, even base 1. However, there is
only one infinite number is base 1: . . .111111
Nigel Cole * 10002...@compuserve.com
-------------------------------------------------------------
Newsgroup: sci.math
From: nhc...@zebekia.demon.co.uk (Nigel Cole)
Subject: Re: World's first proof of Fermat's Last Theorem and
References: <2q772k$n...@dartvax.dartmouth.edu>
<768548...@zebekia.demon.co.uk>
Organization:
Lines: 34
Date: Wed, 11 May 1994 07:19:53 +0000
Message-ID: <768664...@zebekia.demon.co.uk>
And while you're thinking of A-C, here are proofs D-F:
D) At each step of the generation process, the number created has a
digital root (corresponding to the number modulo 9). This number cycles
from 1 (for 1) to 0 (for 9) and then back to 1 again. However, most of
the infinite numbers do not have well-defined digital roots (eg.
...1111). Thus, the infinite numbers are not Natural numbers, since the
Natural numbers have digital roots and most of the infinite numbers do
not.
E) Adapting the proof that there are more real numbers than integers. .
. It is possible, conceptually, to list the Natural numbers - each has
a well-defined place on the list (you can always find number n on line
n). Imagine doing this with all the infinite numbers. Now consider the
infinite number produced by taking the first digit of the first entry
on the list, the second digit of the second entry, the third digit of
the third entry, etc. With each digit, add 1 to it, modulo whatever
base you are working in. The resulting number, by construction, cannot
already be in the table, since it will differ in at least one column
from every number in the table. Hence, the infinite numbers is greater
than the number of Natural numbers. Hence, the infinite numbers are not
the Natural numbers.
F) Consider the process you gave for generating the numbers, considered
in terms of your infinite numbers. At each stage, you are generating a
number in the . . .000a family, where a is some sequence of digits. At
no point do you suddenly switch into, say, the . . .111a family, since
at each step, you still have an infinite number of zero digits ahead of
you number.
Nigel Cole * 10002...@compuserve.com
------- end of 1993 posts where Nigel beautifully defines what Finite
means and proves in less than 6 proofs that Naturals = Finite Integers
----------
Archimedes Plutonium
Archimedes...@dartmouth.edu (Archimedes Plutonium) writes:
> F) Consider the process you gave for generating the numbers, considered
> in terms of your infinite numbers. At each stage, you are generating a
> number in the . . .000a family, where a is some sequence of digits. At
> no point do you suddenly switch into, say, the . . .111a family, since
> at each step, you still have an infinite number of zero digits ahead of
> you number.
>
> Nigel Cole * 10002...@compuserve.com
> ------- end of 1993 posts where Nigel beautifully defines what Finite
> means and proves in less than 6 proofs that Naturals = Finite Integers
> ----------
I posted that Nigel Cole did that in 1993, that was an error. Nigel
posted his six beautiful proofs in 1994. Another error on my part I
said "and proves in less than 6 proofs that Naturals = Finite
Integers". That should have read "and proves in no less than, . ."
The above sixth proof that Naturals = Finite Integers is the one that
Princeton Math Dept (combined math and gymnastics department Oct1993)
uses whenever anyone, such as a new Princeton Freshman student decides
to major in math. And, since Andy Panda Wiles feels that he nor his
Teddy Bear Taylor in Cambridge need to prove Naturals = Finite
Integers, then Nigel Cole (probably another Cambridge or Oxford
graduate) gives to the math world the only acceptable and very
beautiful proofs that says for all time
NATURALS = FINITE INTEGERS
and not what A.Plutonium says and has given 7 proofs thereof that
Naturals = Adics = Infinite Integers = Riemannian Geometry
Oh, by the way Nigel. Surely your definition of "finite" distinguishes
say your finite number 99 from the 10-adic of ......00000099r where r
is radix, and your finite number 100. Because, Nigel in your
magnificent and beautifully rosy proofs your definition of finite seems
to have some kind of a logical gap, for if you notice that 99 has two
place values whereas 100 has 3 place values but all of adics have
infinite place value. So , I , guess Wiles and Conway and Cole and
Conrad's and Taylor's definition of Finite Integer will reconcile
finite place value also? In other words finite integer is related to
finite place value , as to your Peano Axiom and your Z-F axiomatics
the successor not only only adds 1 infinitely , which somehow or other
your definition of finiteness will prevent from filling up all
of those place values.
To painful for you to grapple with Messr. Conway and Wiles? Of course
it is, neither one of you had any math genius to ever realize the
failings of pre1993 math.