Goldbach and Euler’s proof is a typical example of what some historians consider a misuse of divergent series , for it starts by assigning a “value” to the harmonic series ∑1/n and proceeds by manipulating it by substraction and replacement of other series until the desired result is reached.
Hi Dr Jaykov Foukzon . Others say that it involved a systematic confusion. Actually, neither is the case. Surprisingly, Euler committed no mistake on the matter. The solution to this puzzle is found buried in history under layers of ambiguous expressions, notations, and changing conventions on the definitions of basic operations. See the link: https://www.jstor.org/stable/27642191
Also, see the following link: https://math.stackexchange.com/questions/763998/eulers-errors/764004
Kindly, see the link: https://math.stackexchange.com/questions/2984910/how-is-eulers-formula-wrong
Dear Aref Wazwaz many thanks for important refereces
https://www.researchgate.net/deref/https%3A%2F%2Fwww.jstor.org%2Fstable%2F27642191
Agree with Aref Wazwaz
Incidentally, it is "normal" for pure mathematicians to bury proofs under mountains of guesswork and ambiguous expressions.
Thank you so much dear Gioacchino de Candia for your appreciation.
Wish you the best always.
Incidentally, it is "normal" for pure mathematicians to bury proofs under mountains of guesswork and ambiguous expressions.
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I agree with Gioacchino de Candia.
And the most striking example of such behavior is the so-called "Great" Fermat's theorem, FLT. Almost 400 years of fun or tragic searches. Searches for "nothing".
But, in fact, this is nothing more than a lemma under the really Great Pythagorean Theorem, PT. This lemma can be easily proved using no more than a couple of lines on a sheet of standard paper.
Euler time was that of mostly exploration, prior to another of rigour.
So yes, some things could be wrong, or deduced improperly.
to Juan Weisz
Euler's main mistake is to deny the concept of the limit.
The Euler approach was formalized by Robinson and as you know it is equivalent to the usual analysis using the notion of the limit of a countable sequence.
Robinson deliberately misled many by calling this approuch non-standard analysis
My Final Revised Version dated August 5, 2022, is the one and only valid elementary proof for Fermat's Last Theorem, which has been uploaded in ResearchGate as preprint ( enclosed herewith), which is to be published in a journal.
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