Re: Fermat extended - Only for mathemathicians
- Posted by encephalon1 at msn.com Aug 15, 2002
- 505 views
------=_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() > > > > FREE MSN Explorer download : http://explorer.msn.com ------=_NextPart_001_0000_01C24414.EE669490 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= o: encephalon1 at msn.com<BR><BR>EASY UNSUBSCRIBE click here: http://topica.= com/u/?b1dd66.b2PHWj<BR>Or send an email to: EUforum-unsubscribe at topica.c= om<BR><BR>T O P I C A -- Register now to manage your mail!<BR>http://www.= topica.com/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= rom the Web. FREE MSN Explorer download : <a href=3D'http://explorer.msn= .com'>http://explorer.msn.com</a><br></p> ------=_NextPart_001_0000_01C24414.EE669490--