
Re: Comparing the Wiles flaw of FLT with the Euler flaw of exp3 FLT, a huge gap error in both offerings
Posted:
Sep 3, 2017 10:53 PM


On 09/03/2017 10:45 PM, David Bernier wrote: > On 09/03/2017 03:51 PM, David Bernier wrote: >> On 09/03/2017 07:12 AM, David Bernier wrote: >>> On 09/02/2017 04:01 PM, Nobody wrote: >>>> On 9/2/2017 6:12 PM, Archimedes Plutonium wrote: >>>>> First the Euler big huge mistake in FLT >>>> ... >>>>> Now it is horribly amazing that from 1770 to 2016, noone in the math >>>>> community had enough logical brains to realize the gaping hole in >>>>> Euler's alleged proof of case exponent 3. >>>>> >>>>> Laughable, how crippled of logic are most math professors. Laughable >>>>> how they would accept the above proof of exponent 3. >>>>> >>>>> Why so harsh on Euler? What was Euler's mistake? And am I justified >>>>> in being so harsh? >>>>> >>>>> Yes, I am justified in being so harsh, because what Euler missed is >>>>> that his so called proof covers only the fact of when A, B, C are two >>>>> odds with one even. Euler completely loses the fact that a solution >>>>> in exponent 3 can occur when A, B, C are all three even numbers. >>>> >>>> The case of A, B, C being all even is not interesting. If A, B, C >>>> are all even, the equation can be rewritten with A=2D, B=2E, C=2F >>>> with D, E, F all integral because A, B, C are all even. >>>> We now have an equation (2D)^3+(2E)^3=(2F)^3. >>>> Or, 8D^3+8E^3=8F^3. >>>> Factor out 8 and we now have a new equation of the form D^3+E^3=F^3. >>>> The >>>> exact same problem. However, there is a difference. Either D, E, F are >>>> not all even or they are. If they aren't, the all even problem just >>>> reduces to a not all even problem. If they are all even, simply >>>> repeat the divide by 8 step, repeatedly if necessary, until one or >>>> more are not even. >>>> >>>> Just like with Pythagorean triangles. The first is the 3,4,5 >>>> triangle. There is also a 6,8,10 triangle, but all it is is the >>>> 3,4,5 triangle scaled up. >>> >>> Typical of AP: little math intuition, overconfidence, and hubris. >>> >>> Euler : >>> >>> (a) Proved 2^32 + 1 composite, disproving Fermat on Fermat primes. >>> >>> (b) Summed the series: 1/1^2 + 1/2^2 + 1/3^2 + ... = pi^2/6 . >>> >>> (c) Solved the Konigsberg Bridge Problem. >> >> After looking things up, I'd say the 1770 proof/argument in Euler's >> Algebra had >> a gap, a truetoform small gap, and it was noticed by a few. >> As things standed, infinite descent was in question, unless the gap >> was resolved. >> >> >> This was discussed in 2000 in sci.math : >> >> "[HM] Fermat's Last Theorem (was: alhandasah)" >> by Franz Lemmermeyer, at: >> >> < http://mathforum.org/kb/message.jspa?messageID=1180657 > . >> >> >> >> Lemmermeyer wrote: >> >> "BTW, in >> Supplementum quorundam theorematum arithmeticorum >> quae in nonnullis demonstrationibus supponuntur, >> Novi. Comm. Acad. Sci. Petrop. 8 (1760/61), 1763, 105128; >> Opera Omnia I  2, 556575 >> Euler closes the gap in a proof of FLT for n = 3 that has apparently >> never appeared in print (is this correct? My Latin [...] " >> >> >> Emili Bifet wrote: >> >> "Dear All, >> This "Supplementum ..." is available online at BnF: >> http://gallica.bnf.fr/scripts/get_page.exe?F=PDF&O=006952&E=596&N=20&CD=1 >> (20 pages in Latin, ~1 MB) " >> >> >> >> Paulo Ribenboim also mentions Euler's 1760 "Supplementum quorandum...." >> in the References for Lecture III, "B.K." Before Kummer, >> at: >> >> "13 Lectures on Fermat's Last Theorem" ( Ribenboim ): >> >> < https://books.google.ca/books?id=c6TTBwAAQBAJ > . >> > > I was trying to figure out how much Euler proved, and how much or > what he assumed without proof, if anything, in his > 1760 communication to the St. Petersburg Academy. > > It's in Latin, which I can't read, so I'm not sure. > > The Latin expression: > > De numeris formae aa + 3bb > > appears at the top of this work as it appears in > reprinted in "Euler's works" under vatious editions. > > David Bernier
The Supplementum of 1760 is also known as E272:
http://eulerarchive.maa.org/docs/originals/E272.pdf

