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, no-one 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 true-to-form 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: al-handasah)" > 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, 105--128; > Opera Omnia I - 2, 556--575 > 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 on-line 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.