Fermat's Last Theorem simple proof impossible?
in [Math]
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From: DRMARJOHN on 1 May 2010 14:26 > > >> that the "claim" was a personal note made on the > margin of his copy of > >> the book, a note that he probably never ima > > As Weil notes, there is evidence that Fermat > t conjectured > the general case early in his mathematical career > r (one can't > be sure since there is no way to date Fermat's > s marginal notes > in his copy of Bachet's Diophantus). Never again > n does Fermat > mention the general case in any of his writings -- > - only the > special cases for exponents 3 and 4 (on several > l occasions). > This seems to indicate that Fermat later realized > d that > his methods did not apply to the general case. > . Moreover, > we now know that to be the case -- his techniques > s are all > special cases of general results on conic and > d elliptic curves; > these techniques do not generalize to higher > r exponents. > THERE IS A GENERAL AND SIMPLE APPROACH THAT INCLUDES ALL EXPONENTS. USING An + Bn = 1, An & Bn as decimals, starting as .5 + .5 = 1 : SINCE .5000 HAS IRRATIONAL ROOTS, THERE IS A PAIR OF ROOTS THAT ARE MORE THAN AND LESS THAN THE IRRATIONAL ROOT AT ANY EXPONENT, THE DIFFERENCE BETWEEN THE PAIR BEING ONE DIGIT AT THE END. ONE WILL END IN A EVEN DIGIT, THE OTHER IN AN ODD DIGIT. AN ODD AND AN EVEN CAN NEVER ADD TO the EVEN OF Cn (1). > See my 2 posts in [1] for more, including quotes and > references. > > Keep in mind that the amount of number theory known > by Fermat > and his contemporaries is _far_ less than one obtains > nowadays > in a first course in number theory. > > --Bill Dubuque > > [1] > http://google.com/groups?threadm=y8zk733ifwb.fsf%40nes > tle.ai.mit.edu
From: DRMARJOHN on 1 May 2010 14:37 > Why not? I thought Wiles' proof of FLT had a flaw but > I was not right. I have just found an elementary > proof of the following theorem: > For every integer n >= 3, the only rational solutions > of the curve C given by > x^n + y^n - 1 = 0 > are the points (1, 0), (0, 1) if n is odd and (-1, > 0), (0, -1), (1, 0), (0, 1) if n even. I have been interested in the odd and even aspect of FLT , and when Cn = 1. May I have your reference? DRMARJOHN
From: J. Clarke on 1 May 2010 18:58 On 5/1/2010 6:26 PM, DRMARJOHN wrote: >> >>>> that the "claim" was a personal note made on the >> margin of his copy of >>>> the book, a note that he probably never ima >> >> As Weil notes, there is evidence that Fermat >> t conjectured >> the general case early in his mathematical career >> r (one can't >> be sure since there is no way to date Fermat's >> s marginal notes >> in his copy of Bachet's Diophantus). Never again >> n does Fermat >> mention the general case in any of his writings -- >> - only the >> special cases for exponents 3 and 4 (on several >> l occasions). >> This seems to indicate that Fermat later realized >> d that >> his methods did not apply to the general case. >> . Moreover, >> we now know that to be the case -- his techniques >> s are all >> special cases of general results on conic and >> d elliptic curves; >> these techniques do not generalize to higher >> r exponents. >> > THERE IS A GENERAL AND SIMPLE APPROACH THAT INCLUDES ALL EXPONENTS. USING An + Bn = 1, An& Bn as decimals, starting as .5 + .5 = 1 : SINCE .5000 HAS IRRATIONAL ROOTS, THERE IS A PAIR OF ROOTS THAT ARE MORE THAN AND LESS THAN THE IRRATIONAL ROOT AT ANY EXPONENT, THE DIFFERENCE BETWEEN THE PAIR BEING ONE DIGIT AT THE END. ONE WILL END IN A EVEN DIGIT, THE OTHER IN AN ODD DIGIT. AN ODD AND AN EVEN CAN NEVER ADD TO the EVEN OF Cn (1). Fine, publish your proof and collect your Fields Medal. >> See my 2 posts in [1] for more, including quotes and >> references. >> >> Keep in mind that the amount of number theory known >> by Fermat >> and his contemporaries is _far_ less than one obtains >> nowadays >> in a first course in number theory. >> >> --Bill Dubuque >> >> [1] >> http://google.com/groups?threadm=y8zk733ifwb.fsf%40nes >> tle.ai.mit.edu
From: spudnik on 1 May 2010 21:39 with only the "trivial" solutions on the curves o'Fermatttt, it sounds like a "necessary but insufficient" proof; PdF certainly could have done it. > I have been interested in the odd and even aspect of FLT , and when Cn = 1. May I have your reference? DRMARJOHN thus: so, your coinage of pi(a,b) is the same as pi(b) - pi(a); now, can you say thr proof as a wordprolemmum? --Light: A History! http://wlym.takeTHEgoogolOUT.com
From: DRMARJOHN on 3 May 2010 12:12
> On 5/1/2010 6:26 PM, DRMARJOHN wrote: > >> > >>>> that the "claim" was a personal note made on the > >> margin of his copy of > >>>> the book, a note that he probably never ima > >> > >> As Weil notes, there is evidence that Fermat > >> t conjectured > >> the general case early in his mathematical career > >> r (one can't > >> be sure since there is no way to date Fermat's > >> s marginal notes > >> in his copy of Bachet's Diophantus). Never again > >> n does Fermat > >> mention the general case in any of his writings -- > >> - only the > >> special cases for exponents 3 and 4 (on several > >> l occasions). > >> This seems to indicate that Fermat later realized > >> d that > >> his methods did not apply to the general case. > >> . Moreover, > >> we now know that to be the case -- his techniques > >> s are all > >> special cases of general results on conic and > >> d elliptic curves; > >> these techniques do not generalize to higher > >> r exponents. > >> > > THERE IS A GENERAL AND SIMPLE APPROACH THAT > INCLUDES ALL EXPONENTS. USING An + Bn = 1, An & Bn as > decimals, starting as .5 + .5 = 1 : SINCE .5000 HAS > IRRATIONAL ROOTS, THERE IS A PAIR OF ROOTS THAT ARE > MORE THAN AND LESS THAN THE IRRATIONAL ROOT AT ANY > EXPONENT, THE DIFFERENCE BETWEEN THE PAIR BEING ONE > DIGIT AT THE END. ONE WILL END IN A EVEN DIGIT, THE > OTHER IN AN ODD DIGIT. AN ODD AND AN EVEN CAN NEVER > ADD TO the EVEN OF Cn (1). > > Fine, publish your proof and collect your Fields > Medal. > I AM NOT A MATHEMATICIAN SO NO JOURNAL WILL PUBLISH MY SPECULATIONS. IT IS HERE IN THIS FORUM THAT THESE IDEAS CAN BE EVALUATED. TO CONTINUE: THERE IS AN "A" AND A "B" THAT IS THE FIRST POSSIBLE SET WITH RATIONAL ROOTS MORE THAN AND LESS THAN THE IRRATIONAL ROOT OF .5000. THERE IS A SECOND SET WITH RATIONAL ROOTS THAT IS ABOVE AND BELOW THE ROOT OF .5000 THAT IS SEPARATE BY THREE. THERE IS A THIRD SET THAT IS SEPARATE BY 5, THERE IS A 4TH SET....THERE IS AN NTH SET. ALL THESE SETS REPRESENT THE TOTAL DOMAIN OF RATIONAL NUMBERS. THERE IS A SIMPLE COMPUTATIONAL REASON THAT SUCH An AND Bn CAN NEVER ADD TO I.000. ALL THE PAIRS OF A AND B ARE PAIRS OF EVEN AND ODD. THE EXPONENTIAL MULTIPLICATION OF AN EVEN DIGIT ALWAYS GIVES AN EVEN NUMBER, OF AN ODD ALWAYS YIELDS AN ODD. AN EVEN AND AN ODD NEVER CAN ADD TO A Cn OF 1.000 THAT IS EVEN. I AM ONLY PRESENTING THIS SIMPLE APPROACH TO ILLUSTRATE THE PROCESS OF FLT. IT DOES HAVE THE SIMPLICITY THAT FERMAT COULD ENTERTAIN IN HIS ERA. AND I UNDERSTAND THAT HE RARELY GAVE AWAY THE BASIS FOR HIS THINKING. PLEASE CONTINUE THIS DISCUSSION. MARTIN JOHNSON > >> See my 2 posts in [1] for more, including quotes > and > >> references. > >> > >> Keep in mind that the amount of number theory > known > >> by Fermat > >> and his contemporaries is _far_ less than one > obtains > >> nowadays > >> in a first course in number theory. > >> > >> --Bill Dubuque > >> > >> [1] > >> > http://google.com/groups?threadm=y8zk733ifwb.fsf%40nes > >> tle.ai.mit.edu > |