Re: Fermat extended - Only for mathemathicians
- Posted by encephalon1 at msn.com
Aug 15, 2002
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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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<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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