1. Fermat extended - Only for mathemathicians
- Posted by rforno at tutopia.com
Aug 14, 2002
Dear EUphorians:
Apparently, someone proved the last theorem by Fermat to be true.
I don't know if the following extension to Fermat's theorem (or conjecture)
has ever be posed by someone, but here it is, with a program that tries to
find a counterexample. I've found no one yet.
Comments are welcome.
-- Trying to find a counterexample for the "extended-Fermat conjecture",
that
-- x[1]^p+x[2]^p...+x[n]^p = z^p, for x[i] > 0, 1 < n < p has no integer
-- solutions.
-- Author R. M. Forno - Version 1.0 - 2002/08/13
constant COMPL = 30 -- Start with numbers somewhat big
sequence top -- The elements
procedure verify(integer n, integer p) -- Verify conjecture
atom root, sum
integer r
sum = 0
for i = 1 to n do -- Always perform the sum to avoid rounding errors
sum += power(top[i], p)
end for
root = power(sum, 1 / p)
r = floor(root + 0.5) -- Beware of rounding errors
if power(r, p) = sum then -- Show results... some day
printf(1, "Power: %d Left: %f Right:", {p, root})
for i = 1 to n do
printf(1, " %d", top[i])
end for
puts(1, '\n')
end if
end procedure
procedure fermat()
integer p, k, i, r
p = 2
while p <= 20 do
p += 1
printf(1, "Testing exponent %d\n", p)
r = p - 1
for n = 2 to r - 1 do -- Previous powers
top = repeat(r + COMPL, r)
verify (n, n + 1)
i = n
while i > 1 do
while top[i] > 1 do
top[i] -= 1
k = top[i]
while i < n do
i += 1
top[i] = k -- Avoid repeating previous tests
end while
verify(n, n + 1)
end while
i -= 1
end while
end for
for n = 2 to r do -- Present power
top = repeat(r + COMPL, r)
verify (n, p)
i = n
while i do
while top[i] > 1 do
top[i] -= 1
k = top[i]
while i < n do
i += 1
top[i] = k -- Avoid repeating previous tests
end while
verify(n, p)
end while
i -= 1
end while
end for
end while
end procedure
fermat()
2. Re: Fermat extended - Only for mathemathicians
- Posted by rswiston at hotmail.com
Aug 14, 2002
I think your information is incorrect. Femat's theorem is only true since
no one can find a value of p to make it false. (Remember theorem means a
statement yet to be proven false). To prove the theorem, one must provide
proof that the "conjecture" is true for all values of p (not simply a few
selected ones... and infinity is a pretty big number!). Since this is
nearly inpossible (I say nearly since super computers can do marvelous
things these days) no one has yet to prove (or disprove) the theorem. Your
program can produce sums given a provided input from the variables, but will
still not "prove" the theorem. It is simply looking for a value that will
prove the theorem false (which may well happen if allowed to run long
enough... in a loop that doesn't end until the proof has been found invalid)
and therefore does not prove the theorem. (AUUGHH high school geometry
rears its ugly head!). Unfortunately, computer programs are designed around
numbers either set or inputed from a loop or user input. They are not
designed to act upon an unknown (do not confuse the term variable we
mathmeticians use with the same term computer programmers use).
-Robert
----- Original Message -----
From: <rforno at tutopia.com>
To: "EUforum" <EUforum at topica.com>
Sent: Tuesday, August 13, 2002 10:08 PM
Subject: Fermat extended - Only for mathemathicians
>
> Dear EUphorians:
> Apparently, someone proved the last theorem by Fermat to be true.
> I don't know if the following extension to Fermat's theorem (or
conjecture)
> has ever be posed by someone, but here it is, with a program that tries to
> find a counterexample. I've found no one yet.
> Comments are welcome.
>
> -- Trying to find a counterexample for the "extended-Fermat conjecture",
> that
> -- x[1]^p+x[2]^p...+x[n]^p = z^p, for x[i] > 0, 1 < n < p has no integer
> -- solutions.
> -- Author R. M. Forno - Version 1.0 - 2002/08/13
>
> constant COMPL = 30 -- Start with numbers somewhat big
> sequence top -- The elements
>
> procedure verify(integer n, integer p) -- Verify conjecture
> atom root, sum
> integer r
> sum = 0
> for i = 1 to n do -- Always perform the sum to avoid rounding errors
> sum += power(top[i], p)
> end for
> root = power(sum, 1 / p)
> r = floor(root + 0.5) -- Beware of rounding errors
> if power(r, p) = sum then -- Show results... some day
> printf(1, "Power: %d Left: %f Right:", {p, root})
> for i = 1 to n do
> printf(1, " %d", top[i])
> end for
> puts(1, '\n')
> end if
> end procedure
>
> procedure fermat()
> integer p, k, i, r
> p = 2
> while p <= 20 do
> p += 1
> printf(1, "Testing exponent %d\n", p)
> r = p - 1
> for n = 2 to r - 1 do -- Previous powers
> top = repeat(r + COMPL, r)
> verify (n, n + 1)
> i = n
> while i > 1 do
> while top[i] > 1 do
> top[i] -= 1
> k = top[i]
> while i < n do
> i += 1
> top[i] = k -- Avoid repeating previous tests
> end while
> verify(n, n + 1)
> end while
> i -= 1
> end while
> end for
> for n = 2 to r do -- Present power
> top = repeat(r + COMPL, r)
> verify (n, p)
> i = n
> while i do
> while top[i] > 1 do
> top[i] -= 1
> k = top[i]
> while i < n do
> i += 1
> top[i] = k -- Avoid repeating previous tests
> end while
> verify(n, p)
> end while
> i -= 1
> end while
> end for
> end while
> end procedure
>
> fermat()
>
>
>
>
3. Re: Fermat extended - Only for mathemathicians
- Posted by encephalon1 at msn.com
Aug 15, 2002
------=_NextPart_001_0000_01C24414.EE669490
actually, there is a rather large book that details the proof ( that is, =
details of the steps taken to solve the problem ) of fermats theorem that=
was written (I believe) in the nineties--I've never read the work myself=
, but I seen it referenced quite a few times. If I can, I'll try to find=
the title.
~Nathan
----- Original Message -----
From: rforno at tutopia.com
Subject: RE: Fermat extended - Only for mathemathicians
Apparently you did not read the caption at the top of the program, or my
English is very bad.
I know that this program cannot prove the conjecture to be true, but it c=
an
prove it to be false.
Moreover, I was interested in knowing if someone posed the same conjectur=
e
that I call "extended Fermat theorem".
About the original conjecture, I remember having heard that someone final=
ly
proved it to be true, not by means of a computer program but using advanc=
ed
math theory.
----- Original Message -----
From: <rswiston at hotmail.com>
To: EUforum <EUforum at topica.com>
Sent: Thursday, August 15, 2002 1:23 AM
Subject: Re: Fermat extended - Only for mathemathicians
>
> I think your information is incorrect. Femat's theorem is only true si=
nce
> no one can find a value of p to make it false. (Remember theorem means=
a
> statement yet to be proven false). To prove the theorem, one must prov=
ide
> proof that the "conjecture" is true for all values of p (not simply a f=
ew
> selected ones... and infinity is a pretty big number!). Since this is
> nearly inpossible (I say nearly since super computers can do marvelous
> things these days) no one has yet to prove (or disprove) the theorem.
Your
> program can produce sums given a provided input from the variables, but
will
> still not "prove" the theorem. It is simply looking for a value that w=
ill
> prove the theorem false (which may well happen if allowed to run long
> enough... in a loop that doesn't end until the proof has been found
invalid)
> and therefore does not prove the theorem. (AUUGHH high school geometry
> rears its ugly head!). Unfortunately, computer programs are designed
around
> numbers either set or inputed from a loop or user input. They are not
> designed to act upon an unknown (do not confuse the term variable we
> mathmeticians use with the same term computer programmers use).
>
> -Robert
>
> ----- Original Message -----
> From: <rforno at tutopia.com>
> To: "EUforum" <EUforum at topica.com>
> Sent: Tuesday, August 13, 2002 10:08 PM
> Subject: Fermat extended - Only for mathemathicians
>
>
> > Dear EUphorians:
> > Apparently, someone proved the last theorem by Fermat to be true.
> > I don't know if the following extension to Fermat's theorem (or
> conjecture)
> > has ever be posed by someone, but here it is, with a program that tri=
es
to
> > find a counterexample. I've found no one yet.
> > Comments are welcome.
> >
> > -- Trying to find a counterexample for the "extended-Fermat conjectur=
e",
> > that
> > -- x[1]^p+x[2]^p...+x[n]^p =3D z^p, for x[i] > 0, 1 < n < p has no in=
teger
> > -- solutions.
> > -- Author R. M. Forno - Version 1.0 - 2002/08/13
> >
> > constant COMPL =3D 30 -- Start with numbers somewhat big
> > sequence top -- The elements
> >
> > procedure verify(integer n, integer p) -- Verify conjecture
> > atom root, sum
> > integer r
> > sum =3D 0
> > for i =3D 1 to n do -- Always perform the sum to avoid rounding e=
rrors
> > sum +=3D power(top[i], p)
> > end for
> > root =3D power(sum, 1 / p)
> > r =3D floor(root + 0.5) -- Beware of rounding errors
> > if power(r, p) =3D sum then -- Show results... some day
> > printf(1, "Power: %d Left: %f Right:", {p, root})
> > for i =3D 1 to n do
> > printf(1, " %d", top[i])
> > end for
> > puts(1, '\n')
> > end if
> > end procedure
> >
> > procedure fermat()
> > integer p, k, i, r
> > p =3D 2
> > while p <=3D 20 do
> > p +=3D 1
> > printf(1, "Testing exponent %d\n", p)
> > r =3D p - 1
> > for n =3D 2 to r - 1 do -- Previous powers
> > top =3D repeat(r + COMPL, r)
> > verify (n, n + 1)
> > i =3D n
> > while i > 1 do
> > while top[i] > 1 do
> > top[i] -=3D 1
> > k =3D top[i]
> > while i < n do
> > i +=3D 1
> > top[i] =3D k -- Avoid repeating previous tes=
ts
> > end while
> > verify(n, n + 1)
> > end while
> > i -=3D 1
> > end while
> > end for
> > for n =3D 2 to r do -- Present power
> > top =3D repeat(r + COMPL, r)
> > verify (n, p)
> > i =3D n
> > while i do
> > while top[i] > 1 do
> > top[i] -=3D 1
> > k =3D top[i]
> > while i < n do
> > i +=3D 1
> > top[i] =3D k -- Avoid repeating previous tes=
ts
> > end while
> > verify(n, p)
> > end while
> > i -=3D 1
> > end while
> > end for
> > end while
> > end procedure
> >
> > fermat()
> >
> >
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Content-Type: text/html; charset="iso-8859-1"
Content-Transfer-Encoding: quoted-printable
<HTML><BODY STYLE=3D"font:10pt verdana; border:none;"><DIV> </DIV> <=
DIV>actually, there is a rather large book that details the proof ( =
that is, details of the steps taken to solve the problem ) of fermats the=
orem that was written (I believe) in the nineties--I've never read the wo=
rk myself, but I seen it referenced quite a few times. If I can, I'=
ll try to find the title.</DIV> <DIV> </DIV> <DIV>~Nathan</DIV> <BLO=
CKQUOTE style=3D"PADDING-RIGHT: 0px; PADDING-LEFT: 5px; MARGIN-LEFT: 5px;=
BORDER-LEFT: #000000 2px solid; MARGIN-RIGHT: 0px"> <DIV style=3D"FONT: =
10pt Arial">----- Original Message -----</DIV> <DIV style=3D"BACKGROUND: =
#e4e4e4; FONT: 10pt Arial; COLOR: black"><B>From:</B> rforno at tutopia.com<=
/DIV> <DIV style=3D"FONT: 10pt Arial"><B>Sent:</B> Thursday, August 15, 2=
002 12:44 AM</DIV> <DIV style=3D"FONT: 10pt Arial"><B>To:</B> EUforum</DI=
V> <DIV style=3D"FONT: 10pt Arial"><B>Subject:</B> RE: Fermat extended - =
Only for mathemathicians</DIV> <DIV> </DIV>=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D The Euphoria Mailing List =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D =
<BR><BR>Apparently you did not read the caption at the top of the program=
, or my<BR>English is very bad.<BR>I know that this program cannot prove =
the conjecture to be true, but it can<BR>prove it to be false.<BR>Moreove=
r, I was interested in knowing if someone posed the same conjecture<BR>th=
at I call "extended Fermat theorem".<BR>About the original conjecture, I =
remember having heard that someone finally<BR>proved it to be true, not b=
y means of a computer program but using advanced<BR>math theory.<BR>-----=
Original Message -----<BR>From: <rswiston at hotmail.com><BR>To: EUfo=
rum <EUforum at topica.com><BR>Sent: Thursday, August 15, 2002 1:23 AM=
<BR>Subject: Re: Fermat extended - Only for mathemathicians<BR><BR><BR>&g=
t;<BR>> I think your information is incorrect. Femat's theorem i=
s only true since<BR>> no one can find a value of p to make it false.&=
nbsp; (Remember theorem means a<BR>> statement yet to be proven false)=
. To prove the theorem, one must provide<BR>> proof that the "co=
njecture" is true for all values of p (not simply a few<BR>> selected =
ones... and infinity is a pretty big number!). Since this is<BR>>=
; nearly inpossible (I say nearly since super computers can do marvelous<=
BR>> things these days) no one has yet to prove (or disprove) the theo=
rem.<BR>Your<BR>> program can produce sums given a provided input from=
the variables, but<BR>will<BR>> still not "prove" the theorem. =
It is simply looking for a value that will<BR>> prove the theorem fals=
e (which may well happen if allowed to run long<BR>> enough... in a lo=
op that doesn't end until the proof has been found<BR>invalid)<BR>> an=
d therefore does not prove the theorem. (AUUGHH high school geometr=
y<BR>> rears its ugly head!). Unfortunately, computer programs a=
re designed<BR>around<BR>> numbers either set or inputed from a loop o=
r user input. They are not<BR>> designed to act upon an unknown =
(do not confuse the term variable we<BR>> mathmeticians use with the s=
ame term computer programmers use).<BR>><BR>> -Robert<BR>><BR>&g=
t; ----- Original Message -----<BR>> From: <rforno at tutopia.com><=
BR>> To: "EUforum" <EUforum at topica.com><BR>> Sent: Tuesday, A=
ugust 13, 2002 10:08 PM<BR>> Subject: Fermat extended - Only for mathe=
mathicians<BR>><BR>><BR>> > Dear EUphorians:<BR>> > App=
arently, someone proved the last theorem by Fermat to be true.<BR>> &g=
t; I don't know if the following extension to Fermat's theorem (or<BR>>=
; conjecture)<BR>> > has ever be posed by someone, but here it is, =
with a program that tries<BR>to<BR>> > find a counterexample. I've =
found no one yet.<BR>> > Comments are welcome.<BR>> ><BR>>=
> -- Trying to find a counterexample for the "extended-Fermat conject=
ure",<BR>> > that<BR>> > -- x[1]^p+x[2]^p...+x[n]^p =3D z^p, =
for x[i] > 0, 1 < n < p has no integer<BR>> > -- solutions=
.<BR>> > -- Author R. M. Forno - Version 1.0 - 2002/08/13<BR>> &=
gt;<BR>> > constant COMPL =3D 30 -- Start with numbers somewhat big=
<BR>> > sequence top -- The elements<BR>> ><BR>> > proc=
edure verify(integer n, integer p) -- Verify conjecture<BR>> > =
; atom root, sum<BR>> > i=
nteger r<BR>> > sum =3D 0<BR>> > =
; for i =3D 1 to n do -- Always perform the sum to avoi=
d rounding errors<BR>> > &=
nbsp; sum +=3D power(top[i], p)<BR>> > end =
for<BR>> > root =3D power(sum, 1 / p)<BR>&g=
t; > r =3D floor(root + 0.5) -- Beware of roun=
ding errors<BR>> > if power(r, p) =3D sum t=
hen -- Show results... some day<BR>> >  =
; printf(1, "Power: %d Left: %f Right:", {p, root})<BR>=
> > for i =3D 1 to =
n do<BR>> > &n=
bsp; printf(1, " %d", top[i])<BR>> > &=
nbsp; end for<BR>> >  =
; puts(1, '\n')<BR>> > &nb=
sp; end if<BR>> > end procedure<BR>> ><BR>> > procedure=
fermat()<BR>> > integer p, k, i, r<BR>>=
> p =3D 2<BR>> >  =
; while p <=3D 20 do<BR>> > &=
nbsp; p +=3D 1<BR>> > &nbs=
p; printf(1, "Testing exponent %d\n", p)<BR>> > &n=
bsp; r =3D p - 1<BR>> > &n=
bsp; for n =3D 2 to r - 1 do -- Previ=
ous powers<BR>> > &n=
bsp; top =3D repeat(r + COMPL, r)<BR>> > &nb=
sp; verify (n=
, n + 1)<BR>> > &nbs=
p; i =3D n<BR>> > &n=
bsp; while i > 1 do<BR>> >&n=
bsp; &nb=
sp; while top[i] > 1 do<BR>> > &nbs=
p;  =
; top[i] -=3D 1<BR>> > &nb=
sp; &nbs=
p; k =3D top[i]<BR>> > &nb=
sp; &nbs=
p; while i < n do<BR>> > &nbs=
p;  =
; i +=3D 1<BR=
>> > &nb=
sp; &nbs=
p; top[i] =3D k -- Avoid repeating previous tests<BR>> >=
; =
end while<BR>> >&n=
bsp; &nb=
sp; verify(n, n + 1)<BR>> &g=
t;  =
; end while<BR>> > &=
nbsp; i=
-=3D 1<BR>> >  =
; end while<BR>> > &=
nbsp; end for<BR>> >  =
; for n =3D 2 to r do -- Present power<BR>> > &nbs=
p; top =3D re=
peat(r + COMPL, r)<BR>> > =
verify (n, p)<BR>> > &nbs=
p; i =3D n<BR>> =
> &nb=
sp; while i do<BR>> > &nbs=
p; while top[i] > 1 do=
<BR>> > =
top[i] -=3D =
1<BR>> >  =
; k =3D top[i=
]<BR>> >  =
; while i <=
; n do<BR>> > =
&=
nbsp; i +=3D 1<BR>> > &nbs=
p;  =
; top[i] =3D k -- Avoid repeati=
ng previous tests<BR>> > &=
nbsp; &n=
bsp; end while<BR>> > &nbs=
p;  =
; verify(n, p)<BR>> > &nbs=
p; end while<BR>> >=
&=
nbsp; i -=3D 1<BR>> > &nbs=
p; end while<BR>> > =
end for<BR>> > &nbs=
p; end while<BR>> > end procedure<BR>> ><BR>> =
> fermat()<BR>> ><BR>> ><BR>><BR>><BR>><BR><BR>=3D=
=3D^=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D<BR>This email was sent t=
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com/u/?b1dd66.b2PHWj<BR>Or send an email to: EUforum-unsubscribe at topica.c=
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=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
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4. Re: Fermat extended - Only for mathemathicians
Search for the following at Google Groups and you'll come across a thread I
started on the sci.math newsgroup many moons ago. (I would provide the URL
but that extends to several lines!):
crwhite at comp group:sci.math fermat extend
Click to view the complete thread to see some of the replies I got. I know
for a fact that part of that particular thread is missing because one kind
soul provided a counterexample matching x^4 + y^4 + w^4 = z^4, and that's
not there now.
Here's a spooky coincidence that doesn't quite fit your criteria:
3^2 + 4^2 = 5^2
3^3 + 4^3 + 5^3 = 6^3
Carl
5. Re: Fermat extended - Only for mathemathicians
- Posted by encephalon1 at msn.com
Aug 15, 2002
------=_NextPart_001_0000_01C244AF.9A20ED90
hey.. that's the book I was speaking of =3D]
----- Original Message -----
From: munchr at mac.com
Subject: RE: Fermat extended - Only for mathemathicians
=3D=3D=3D=3D=3D=3D=3D=3D =20
Andrew Wiles solved Fermat's Last Theorem. His first solution, announced
on June 23, 1993 at the Isaac Newton College, in Cambridge, was found to
be flawed. He was able to fix it though, and in 1995 his accepted proof =
was
published in the Annals of Mathematics, titled "Modular elliptic curves a=
nd
Fermat's Last Theorem".
Suggested reading, for those who are interested can be found at:
http://www.google.com/search?hl=3Den&lr=3D&ie=3DISO-8859-1&q=3Dandrew+wil=
es
http://www.google.com/search?sourceid=3Dnavclient&q=3Dfermat+last+theorem
Also, check out the book "Fermat's Last Theorem" by Simon Singh, which
details Andrew Wiles achievement, as well as the many attempts by others
to prove the theorem. It can be found at http://www.simonsingh.com/ferma=
t.htm
James Powell
>Apparently you did not read the caption at the top of the program, or my
>English is very bad.
>I know that this program cannot prove the conjecture to be true, but it =
can
>prove it to be false.
>Moreover, I was interested in knowing if someone posed the same conjectu=
re
>that I call "extended Fermat theorem".
>About the original conjecture, I remember having heard that someone fina=
lly
>proved it to be true, not by means of a computer program but using advan=
ced
>math theory.
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D
This email was sent to: encephalon1 at msn.com
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=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3DGet more from the Web.=
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<HTML><BODY STYLE=3D"font:10pt verdana; border:none;"><DIV>hey.. that's t=
he book I was speaking of =3D]</DIV> <DIV> </DIV> <BLOCKQUOTE
style=3D=
"PADDING-RIGHT: 0px; PADDING-LEFT: 5px; MARGIN-LEFT: 5px; BORDER-LEFT: #0=
00000 2px solid; MARGIN-RIGHT: 0px"> <DIV style=3D"FONT: 10pt Arial">----=
- Original Message -----</DIV> <DIV style=3D"BACKGROUND: #e4e4e4; FONT: 1=
0pt Arial; COLOR: black"><B>From:</B> munchr at mac.com</DIV> <DIV style=3D"=
FONT: 10pt Arial"><B>Sent:</B> Thursday, August 15, 2002 1:13 AM</DIV> <D=
IV style=3D"FONT: 10pt Arial"><B>To:</B> EUforum</DIV> <DIV style=3D"FONT=
: 10pt Arial"><B>Subject:</B> RE: Fermat extended - Only for mathemathici=
ans</DIV> <DIV> </DIV>=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D The Eupho=
ria Mailing List =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D <BR><BR>Andrew Wile=
s solved Fermat's Last Theorem. His first solution, announced<BR>on=
June 23, 1993 at the Isaac Newton College, in Cambridge, was found to<BR=
>be flawed. He was able to fix it though, and in 1995 his accepted =
proof was<BR>published in the Annals of Mathematics, titled "Modular elli=
ptic curves and<BR>Fermat's Last Theorem".<BR><BR>Suggested reading, for =
those who are interested can be found at:<BR><BR>http://www.google.com/se=
arch?hl=3Den&lr=3D&ie=3DISO-8859-1&q=3Dandrew+wiles<BR><BR>ht=
tp://www.google.com/search?sourceid=3Dnavclient&q=3Dfermat+last+theor=
em<BR><BR>Also, check out the book "Fermat's Last Theorem" by Simon Singh=
, which<BR>details Andrew Wiles achievement, as well as the many attempts=
by others<BR>to prove the theorem. It can be found at http://www.s=
imonsingh.com/fermat.htm<BR><BR>James Powell<BR><BR>>Apparently you di=
d not read the caption at the top of the program, or my<BR>>English is=
very bad.<BR>>I know that this program cannot prove the conjecture to=
be true, but it can<BR>>prove it to be false.<BR>>Moreover, I was =
interested in knowing if someone posed the same conjecture<BR>>that I =
call "extended Fermat theorem".<BR>>About the original conjecture, I r=
emember having heard that someone finally<BR>>proved it to be true, no=
t by means of a computer program but using advanced<BR>>math theory.<B=
R><BR>=3D=3D^=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D<BR>This email w=
as sent to: encephalon1 at msn.com<BR><BR>EASY UNSUBSCRIBE click here: http:=
//topica.com/u/?b1dd66.b2PHWj<BR>Or send an email to: EUforum-unsubscribe=
@topica.com<BR><BR>T O P I C A -- Register now to manage your mail!<BR>ht=
tp://www.topica.com/partner/tag02/register<BR>=3D=3D^=3D=3D=3D=3D=3D=3D=3D=
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=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
et more from the Web. FREE MSN Explorer download : <a href=3D'http://exp=
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6. Re: Fermat extended - Only for mathemathicians
- Posted by encephalon1 at msn.com
Aug 15, 2002
------=_NextPart_001_0001_01C244B5.695DF5D0
I am slightly confused.. is this "extended conjecture" something that Fer=
mat himself brought into awareness, or is it an idea of your own that ste=
ms from Fermat's Last theorem???? And if it is.. why not call it "rforno=
's theorem?" Maybe I missed something in an earlier post.. what conjectu=
re are you discussing here?
----- Original Message -----
From: rforno at tutopia.com
Subject: RE: Fermat extended - Only for mathemathicians
=3D=3D=3D=3D=3D=3D=3D=3D =20
OK, James, but do you have any information about the "extended conjecture=
"
I
am posing? It is a generalisation of Last Fermat's Theorem. I do not have
the mathematical abilities to try to solve it, so I only tried to find a
counterexample with my program, knowing that if it finds one, the
"Extended
Fermat=B4s Conjecture" will be disproved.
Regards.
----- Original Message -----
From: <munchr at mac.com>
To: EUforum <EUforum at topica.com>
Sent: Thursday, August 15, 2002 2:10 AM
Subject: RE: Fermat extended - Only for mathemathicians
>
> Andrew Wiles solved Fermat's Last Theorem. His first solution,
announced
> on June 23, 1993 at the Isaac Newton College, in Cambridge, was found t=
o
> be flawed. He was able to fix it though, and in 1995 his accepted proo=
f
was
> published in the Annals of Mathematics, titled "Modular elliptic curves
and
> Fermat's Last Theorem".
>
> Suggested reading, for those who are interested can be found at:
>
> http://www.google.com/search?hl=3Den&lr=3D&ie=3DISO-8859-1&q=3Dandrew+w=
iles
>
> http://www.google.com/search?sourceid=3Dnavclient&q=3Dfermat+last+theor=
em
>
> Also, check out the book "Fermat's Last Theorem" by Simon Singh, which
> details Andrew Wiles achievement, as well as the many attempts by other=
s
> to prove the theorem. It can be found at
http://www.simonsingh.com/fermat.htm
>
> James Powell
>
> >Apparently you did not read the caption at the top of the program, or
my
> >English is very bad.
> >I know that this program cannot prove the conjecture to be true, but i=
t
can
> >prove it to be false.
> >Moreover, I was interested in knowing if someone posed the same
conjecture
> >that I call "extended Fermat theorem".
> >About the original conjecture, I remember having heard that someone
finally
> >proved it to be true, not by means of a computer program but using
advanced
> >math theory.
>
>
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D
This email was sent to: encephalon1 at msn.com
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3DGet more from the Web.=
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Content-Type: text/html; charset="iso-8859-1"
Content-Transfer-Encoding: quoted-printable
<HTML><BODY STYLE=3D"font:10pt verdana; border:none;"><DIV> </DIV> <=
DIV>I am slightly confused.. is this "extended conjecture" something that=
Fermat himself brought into awareness, or is it an idea of your own that=
stems from Fermat's Last theorem???? And if it is.. why not call i=
t "rforno's theorem?" Maybe I missed something in an earlier post..=
what conjecture are you discussing here?</DIV> <BLOCKQUOTE style=3D"PADD=
ING-RIGHT: 0px; PADDING-LEFT: 5px; MARGIN-LEFT: 5px; BORDER-LEFT: #000000=
2px solid; MARGIN-RIGHT: 0px"> <DIV style=3D"FONT: 10pt Arial">----- Ori=
ginal Message -----</DIV> <DIV style=3D"BACKGROUND: #e4e4e4; FONT: 10pt A=
rial; COLOR: black"><B>From:</B> rforno at tutopia.com</DIV> <DIV style=3D"F=
ONT: 10pt Arial"><B>Sent:</B> Thursday, August 15, 2002 11:28 PM</DIV> <D=
IV style=3D"FONT: 10pt Arial"><B>To:</B> EUforum</DIV> <DIV style=3D"FONT=
: 10pt Arial"><B>Subject:</B> RE: Fermat extended - Only for mathemathici=
ans</DIV> <DIV> </DIV>=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D The Eupho=
ria Mailing List =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D <BR><BR>OK, James, =
but do you have any information about the "extended conjecture"<BR>I<BR>a=
m posing? It is a generalisation of Last Fermat's Theorem. I do not have<=
BR>the mathematical abilities to try to solve it, so I only tried to find=
a<BR>counterexample with my program, knowing that if it finds one, the<B=
R>"Extended<BR>Fermat=B4s Conjecture" will be disproved.<BR>Regards.<BR>-=
---- Original Message -----<BR>From: <munchr at mac.com><BR>To: EUforu=
m <EUforum at topica.com><BR>Sent: Thursday, August 15, 2002 2:10 AM<B=
R>Subject: RE: Fermat extended - Only for mathemathicians<BR><BR><BR>>=
<BR>> Andrew Wiles solved Fermat's Last Theorem. His first solut=
ion,<BR>announced<BR>> on June 23, 1993 at the Isaac Newton College, i=
n Cambridge, was found to<BR>> be flawed. He was able to fix it =
though, and in 1995 his accepted proof<BR>was<BR>> published in the An=
nals of Mathematics, titled "Modular elliptic curves<BR>and<BR>> Ferma=
t's Last Theorem".<BR>><BR>> Suggested reading, for those who are i=
nterested can be found at:<BR>><BR>> http://www.google.com/search?h=
l=3Den&lr=3D&ie=3DISO-8859-1&q=3Dandrew+wiles<BR>><BR>>=
http://www.google.com/search?sourceid=3Dnavclient&q=3Dfermat+last+th=
eorem<BR>><BR>> Also, check out the book "Fermat's Last Theorem" by=
Simon Singh, which<BR>> details Andrew Wiles achievement, as well as =
the many attempts by others<BR>> to prove the theorem. It can be=
found at<BR>http://www.simonsingh.com/fermat.htm<BR>><BR>> James P=
owell<BR>><BR>> >Apparently you did not read the caption at the =
top of the program, or<BR>my<BR>> >English is very bad.<BR>> >=
;I know that this program cannot prove the conjecture to be true, but it<=
BR>can<BR>> >prove it to be false.<BR>> >Moreover, I was inte=
rested in knowing if someone posed the same<BR>conjecture<BR>> >tha=
t I call "extended Fermat theorem".<BR>> >About the original conjec=
ture, I remember having heard that someone<BR>finally<BR>> >proved =
it to be true, not by means of a computer program but using<BR>advanced<B=
R>> >math theory.<BR>><BR>><BR>><BR>><BR><BR>=3D=3D^=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D<BR>This email was sent to: enceph=
alon1 at msn.com<BR><BR>EASY UNSUBSCRIBE click here: http://topica.com/u/?b1=
dd66.b2PHWj<BR>Or send an email to: EUforum-unsubscribe at topica.com<BR><BR=
>T O P I C A -- Register now to manage your mail!<BR>http://www.topica.co=
m/partner/tag02/register<BR>=3D=3D^=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
eb. FREE MSN Explorer download : <a href=3D'http://explorer.msn.com'>htt=
p://explorer.msn.com</a><br></p>
------=_NextPart_001_0001_01C244B5.695DF5D0--
7. Re: Fermat extended - Only for mathemathicians
- Posted by encephalon1 at msn.com
Aug 17, 2002
------=_NextPart_001_0000_01C2459C.CE0AE370
I see.. This would take quite a bit of processing power. It seems, howe=
ver, that one could conclude that n should be -inf. to inf. Then again, =
I've thought a lot of things in mathematics should be one way.. but they =
aren't :P
I wonder if there are any [ large ] files full of pre-calculated exponent=
s? You could then access it in memory instead of calculating it.. that m=
ight speed things up significantly.
If you can, keep me informed on what you decide to do, if anything.
I think you should call it "R.M. Forno's euphoric extension of Fermat's l=
ast theorem that is no longer a theorem but a law theorem"
or yours could work as well.
----- Original Message -----
From: rforno at tutopia.com
Sent: Saturday, August 17, 2002 1:52 AM
To: EUforum
Subject: RE: Fermat extended - Only for mathemathicians
=3D=3D=3D=3D=3D=3D=3D=3D =20
The conjecture I posed is as follows:
x[1]^p + x[2]^p + ... + x[n]^p =3D z^p, for x[i] > 0 and 1 < n < p, has n=
o integer solutions.
Fermat's Last Theorem is a particular case of it, where n =3D 2.
Apparently my conjecture is not true, as someone in the list said there w=
as a counterexample with n =3D 3 and p =3D 4, but he could not find it in=
the margin of an e-mail. My attempts at that failed up to now.
This conjecture was not posed by Fermat, but by me, who had a little demo=
nstration for it but for which I had no space in the margin of the e-mail=
;). If it is not proved to be false, I will name it "R. M. Forno's non-l=
ast Theorem".
----- Original Message ----- =20
From: encephalon1 at msn.com =20
To: EUforum =20
Sent: Friday, August 16, 2002 12:42 AM
Subject: Re: Fermat extended - Only for mathemathicians
=3D=3D=3D=3D=3D=3D=3D=3D =20
I am slightly confused.. is this "extended conjecture" something that Fer=
mat himself brought into awareness, or is it an idea of your own that ste=
ms from Fermat's Last theorem???? And if it is.. why not call it "rforno=
's theorem?" Maybe I missed something in an earlier post.. what conjectu=
re are you discussing here?
----- Original Message -----
From: rforno at tutopia.com
Sent: Thursday, August 15, 2002 11:28 PM
To: EUforum
Subject: RE: Fermat extended - Only for mathemathicians
=3D=3D=3D=3D=3D=3D=3D=3D =20
OK, James, but do you have any information about the "extended conjecture=
"
I
am posing? It is a generalisation of Last Fermat's Theorem. I do not have
the mathematical abilities to try to solve it, so I only tried to find a
counterexample with my program, knowing that if it finds one, the
"Extended
Fermat=B4s Conjecture" will be disproved.
Regards.
----- Original Message -----
From: <munchr at mac.com>
To: EUforum <EUforum at topica.com>
Sent: Thursday, August 15, 2002 2:10 AM
Subject: RE: Fermat extended - Only for mathemathicians
>
> Andrew Wiles solved Fermat's Last Theorem. His first solution,
announced
> on June 23, 1993 at the Isaac Newton College, in Cambridge, was found t=
o
> be flawed. He was able to fix it though, and in 1995 his accepted proo=
f
was
> published in the Annals of Mathematics, titled "Modular elliptic curves
and
> Fermat's Last Theorem".
>
> Suggested reading, for those who are interested can be found at:
>
> http://www.google.com/search?hl=3Den&lr=3D&ie=3DISO-8859-1&q=3Dandrew+w=
iles
>
> http://www.google.com/search?sourceid=3Dnavclient&q=3Dfermat+last+theor=
em
>
> Also, check out the book "Fermat's Last Theorem" by Simon Singh, which
> details Andrew Wiles achievement, as well as the many attempts by other=
s
> to prove the theorem. It can be found at
http://www.simonsingh.com/fermat.htm
>
> James Powell
>
> >Apparently you did not read the caption at the top of the program, or
my
> >English is very bad.
> >I know that this program cannot prove the conjecture to be true, but i=
t
can
> >prove it to be false.
> >Moreover, I was interested in knowing if someone posed the same
conjecture
> >that I call "extended Fermat theorem".
> >About the original conjecture, I remember having heard that someone
finally
> >proved it to be true, not by means of a computer program but using
advanced
> >math theory.
>
>
=3D=3D^=3D
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D
This email was sent to: encephalon1 at msn.com
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D =3D
Get more from the Web. FREE MSN Explorer download : http://explorer.msn.c=
om
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D
This email was sent to: encephalon1 at msn.com
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3DGet more from the Web.=
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Content-Transfer-Encoding: quoted-printable
<HTML><BODY BGCOLOR=3D"#ffffff" STYLE=3D"font:10pt verdana; border:none;b=
ackground-color:#ffffff; "><DIV>I see.. This would take quite a bit=
of processing power. It seems, however, that one could conclude th=
at n should be -inf. to inf. Then again, I've thought a lot of thin=
gs in mathematics should be one way.. but they aren't :P</DIV> <DIV> =
;</DIV> <DIV>I wonder if there are any [ large ] files full of pre-calcul=
ated exponents? You could then access it in memory instead of calcu=
lating it.. that might speed things up significantly.</DIV> <DIV> </=
DIV> <DIV>If you can, keep me informed on what you decide to do, if anyth=
ing.</DIV> <DIV> </DIV> <DIV>I think you should call it "R.M. Forno'=
s euphoric extension of Fermat's last theorem that is no longer a theorem=
but a law theorem"</DIV> <DIV> </DIV> <DIV>or yours could work as w=
ell.</DIV> <DIV> </DIV> <BLOCKQUOTE style=3D"PADDING-RIGHT: 0px; PAD=
DING-LEFT: 5px; MARGIN-LEFT: 5px; BORDER-LEFT: #000000 2px solid; MARGIN-=
RIGHT: 0px"> <DIV style=3D"FONT: 10pt Arial">----- Original Message -----=
</DIV> <DIV style=3D"BACKGROUND: #e4e4e4; FONT: 10pt Arial; COLOR: black"=
><B>From:</B> rforno at tutopia.com</DIV> <DIV style=3D"FONT: 10pt Arial"><B=
>Sent:</B> Saturday, August 17, 2002 1:52 AM</DIV> <DIV style=3D"FONT: 10=
pt Arial"><B>To:</B> EUforum</DIV> <DIV style=3D"FONT: 10pt Arial"><B>Sub=
ject:</B> RE: Fermat extended - Only for mathemathicians</DIV> <DIV> =
;</DIV> <META content=3D"MSHTML 5.00.2614.3500" name=3DGENERATOR> <STYLE>=
</STYLE> <PRE>=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D The Euphoria Mailing L=
ist =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D </PRE> <DIV><FONT face=3DArial>=
The conjecture I posed is as follows:</FONT></DIV> <DIV><FONT face=3DAria=
l>x[1]^p + x[2]^p + ... + x[n]^p =3D z^p, for x[i] > 0 and 1 <=
n < p, has no integer solutions.</FONT></DIV> <DIV><FONT face=3DArial=
>Fermat's Last Theorem is a particular case of it, where n =3D 2.</FONT><=
/DIV> <DIV><FONT face=3DArial>Apparently my conjecture is not true, as so=
meone in the list said there was a counterexample with n =3D 3 and p =3D =
4, but he could not find it in the margin of an e-mail. My attempts at th=
at failed up to now.</FONT></DIV> <DIV><FONT face=3DArial>This conjecture=
was not posed by Fermat, but by me, who had a little demonstration for i=
t but for which I had no space in the margin of the e-mail ;). If it is n=
ot proved to be false, I will name it "R. M. Forno's non-last Theorem".</=
FONT></DIV> <BLOCKQUOTE style=3D"PADDING-RIGHT: 0px; PADDING-LEFT: 5px; M=
ARGIN-LEFT: 5px; BORDER-LEFT: #000000 2px solid; MARGIN-RIGHT: 0px"> <DIV=
style=3D"FONT: 10pt arial">----- Original Message ----- </DIV> <DIV styl=
e=3D"BACKGROUND: #e4e4e4; FONT: 10pt arial; font-color: black"><B>From:</=
B> <A title=3Dencephalon1 at msn.com href=3D"mailto:encephalon1 at msn.com">enc=
ephalon1 at msn.com</A> </DIV> <DIV style=3D"FONT: 10pt arial"><B>To:</B> <A=
title=3DEUforum at topica.com href=3D"mailto:EUforum at topica.com">EUforum</A=
> </DIV> <DIV style=3D"FONT: 10pt arial"><B>Sent:</B> Friday, August 16, =
2002 12:42 AM</DIV> <DIV style=3D"FONT: 10pt arial"><B>Subject:</B> Re: F=
ermat extended - Only for mathemathicians</DIV>
<DIV><BR></DIV><PRE>=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D The Euphoria Mailing List
=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D </PRE> <DIV> </DIV> <DIV>I am slightly confused.=
. is this "extended conjecture" something that Fermat himself brought int=
o awareness, or is it an idea of your own that stems from Fermat's Last t=
heorem???? And if it is.. why not call it "rforno's theorem?" =
Maybe I missed something in an earlier post.. what conjecture are you di=
scussing here?</DIV> <BLOCKQUOTE style=3D"PADDING-RIGHT: 0px; PADDING-LEF=
T: 5px; MARGIN-LEFT: 5px; BORDER-LEFT: #000000 2px solid; MARGIN-RIGHT: 0=
px"> <DIV style=3D"FONT: 10pt Arial">----- Original Message -----</DIV> <=
DIV style=3D"BACKGROUND: #e4e4e4; FONT: 10pt Arial; COLOR: black"><B>From=
:</B> rforno at tutopia.com</DIV> <DIV style=3D"FONT: 10pt Arial"><B>Sent:</=
B> Thursday, August 15, 2002 11:28 PM</DIV> <DIV style=3D"FONT: 10pt Aria=
l"><B>To:</B> EUforum</DIV> <DIV style=3D"FONT: 10pt Arial"><B>Subject:</=
B> RE: Fermat extended - Only for mathemathicians</DIV> <DIV> </DIV>=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D The Euphoria Mailing List
=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D <BR><BR>OK, James, but do you have any informati=
on about the "extended conjecture"<BR>I<BR>am posing? It is a generalisat=
ion of Last Fermat's Theorem. I do not have<BR>the mathematical abilities=
to try to solve it, so I only tried to find a<BR>counterexample with my =
program, knowing that if it finds one, the<BR>"Extended<BR>Fermat=B4s Con=
jecture" will be disproved.<BR>Regards.<BR>----- Original Message -----<B=
R>From: <munchr at mac.com><BR>To: EUforum <EUforum at topica.com><=
BR>Sent: Thursday, August 15, 2002 2:10 AM<BR>Subject: RE: Fermat extende=
d - Only for mathemathicians<BR><BR><BR>><BR>> Andrew Wiles solved =
Fermat's Last Theorem. His first solution,<BR>announced<BR>> on =
June 23, 1993 at the Isaac Newton College, in Cambridge, was found to<BR>=
> be flawed. He was able to fix it though, and in 1995 his accep=
ted proof<BR>was<BR>> published in the Annals of Mathematics, titled "=
Modular elliptic curves<BR>and<BR>> Fermat's Last Theorem".<BR>><BR=
>> Suggested reading, for those who are interested can be found at:<BR=
>><BR>> http://www.google.com/search?hl=3Den&lr=3D&ie=3DISO=
-8859-1&q=3Dandrew+wiles<BR>><BR>> http://www.google.com/search=
?sourceid=3Dnavclient&q=3Dfermat+last+theorem<BR>><BR>> Also, c=
heck out the book "Fermat's Last Theorem" by Simon Singh, which<BR>> d=
etails Andrew Wiles achievement, as well as the many attempts by others<B=
R>> to prove the theorem. It can be found at<BR>http://www.simon=
singh.com/fermat.htm<BR>><BR>> James Powell<BR>><BR>> >App=
arently you did not read the caption at the top of the program, or<BR>my<=
BR>> >English is very bad.<BR>> >I know that this program can=
not prove the conjecture to be true, but it<BR>can<BR>> >prove it t=
o be false.<BR>> >Moreover, I was interested in knowing if someone =
posed the same<BR>conjecture<BR>> >that I call "extended Fermat the=
orem".<BR>> >About the original conjecture, I remember having heard=
that someone<BR>finally<BR>> >proved it to be true, not by means o=
f a computer program but using<BR>advanced<BR>> >math theory.<BR>&g=
t;<BR>><BR>><BR>><BR><BR>=3D=3D^=3D
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8. Re: Fermat extended - Only for mathemathicians
<html>
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</head>
<body>
<br>
<br>
<a class="moz-txt-link-abbreviated" href="mailto:rforno at tutopia.com">rforno
at tutopia.com</a> wrote:<br>
<blockquote type="cite" cite="mid:0.1700008810.1732954345-1463792126-1029563535
at topica.com">
<meta content="MSHTML 5.00.2614.3500" name="GENERATOR">
<style></style>
<div><font face="Arial"> ... I will name it "R. M. Forno's non-last
Theorem".</font></div>
</blockquote>
<br>
<br>
We hope that is the case! ;)<br>
<br>
9. Re: Fermat extended - Only for mathemathicians
rforno wrote:
> Carl W. wrote:
>
> [IIRC there's] a counterexample matching x^4 + y^4 + w^4 = z^4 ...
>
> > With your data, I wrote a small program in C (for speed) trying to
> > find the counterexample with 3 elements and exponent 4. It failed
> > to encounter any case up to 194 as the maximum value of each element
> > (I imposed this limitation in order to have the numbers in the
> > unsigned long range). I will try it again with higher values using
> > long double numbers, but I don't think I will get some answer. Maybe
> > the only way to find a counterexample will be to use big numbers
> > arithmetic, such as the one available in the ABC language, but it will
> > take a looooong time.
I owe you an apology since I omitted a critical piece of info in my last
post. While I didn't remember the numbers involved for x, y, w and z, I
*did* know that they were quite large - being eight or nine digits apiece. I
should have at least said that... Sorry. :(
Searching for equations in Google is a pain, which is why I didn't do that
before. This time I persevered and turned this up:
http://www.discover.com/may_02/bogglers.html
...and somewhere near the bottom is this beauty:
2682440^4 + 15365639^4 + 18796760^4 = 20615673^4
In case you're wondering this is the *smallest* power 4 counterexample. :)
HTH,
Carl
