Would appear to be a short snippet of the longer and much more interesting.

Yes, indeed it is. I try to link to short videos. Watched this one as well. And watched other topics in math that I had no business watching. Curiosity, I guess. Fascinating fellow.

]]>And Something on prime numbers as a bonus.

Would appear to be a short snippet of the longer and much more interesting.

]]>but I’m not sure now.

I think I found it. I still can’t find the puzzle that he solved, but this was the first video I watched about him.

And Something on prime numbers as a bonus.

We have successfully invested in the S&P 100 using a variant of our usual strategy, viz.,

One all for all, and

All one for one

Athos, Porthos, Aramis and I developed it playing cards while waiting for the next skirmish with the Cardinal.

]]>0) “umbilical stem cell therapy”

2) I’m not in any hurry to jump off of the nearest high altitude cliff nor chasm.

3) https://everipedia.org/wiki/lang_en/Mogwai_(Chinese_culture)

Sincerely,

R2D2 knew exactly what he was doing; C3PO was wrong.

]]>Forgot to mention.

I do believe that the unknown theorem could be related to Quadratic Reciprocity.

Where Quadratic Reciprocity is a subset of a larger theorem.

In theory, if if, then then, Quadratic Reciprocity would not, in theory, help in the odd prime exponent case of FLT.

But, there are interesting patterns in the odd primes.

I have zero reason to believe that Fermat did not find something.

But, I can believe he never wrote about it.

]]>The second sentence was meant to begin “Probably you know that Andrew Wiles’ famous 1995 partial proof of the *Taniyama-Shimura-Weil conjecture* was motivated almost entirely …”

I also misspelled Weil later on — it has only one ‘l’

]]>I see you added another comment, while I was typing 🙂

Probably you know that Andrew Wiles’ famous 1995 partial proof of the was motivated almost entirely by his boyhood dream of proving “Fermat’s Last Theorem.”

Because FLT had long resisted all efforts to crack it, and had (for centuries) no visible connection to any important or general problem, no professional mathematician could devote a major effort to proving (or disproving) FLT without destroying his/her reputation: FLT was a territory strictly reserved for amateurs and crackpots.

It was seeing Gerhard Frey draw his connection between Taniyama-Shimura-Weill (as the conjecture became when it was made more precise) that “closed the circuit” for Wiles: he knew that he *could* work on FLT, because it was now a Very Important Problem. [If I recall correctly, Wiles was in the audience when Frey gave a talk presenting his derivation.]

The part of this that proves the truth of the observation you made in your closing paragraph, is that nobody else in the world made such an intensive attack on the Taniyama-Shimura-Weill conjecture, because of a widely shared belief that finding such a proof was simply beyond current mathematical knowledge.

It was Wiles’s “Don Quixote” determination to solve FLT, that propelled him to do (through about nine years of grueling effort, mostly in secret) what his colleagues believed to be unattainable.

When Gauss wrote those words, he perhaps had in mind the Goldbach conjecture as a salient example. It’s so simple that any child with a few years’ math education can understand it; it seems almost certain to be true; and it has resisted all efforts toward solution.

Gauss was born less than 35 years after the conjecture arrived in mathematical history. At age 277, the conjecture remains a fortress no one has conquered.

]]>Thank you for sharing that longer excerpt, which I have not seen before.

Obviously, the interpretation you put forward is the correct one!

As Gauss observed there, “one cannot predict to what extent one will succeed” … progress in mathematics is notoriously uneven and non-linear. There are enough examples of problems in which progress was stalled for decades, or even centuries, before new discoveries or perspectives led to some major advance.

Even if his “old ideas for a great extension” wouldn’t have been fruitful, it would be fascinating for the history of mathematics to know what lines of investigation he had in mind. I wonder whether his notebooks or correspondence left enough clues, to learn what those old ideas were?

]]>Another remark of Gauss, which illustrates his idea of the character of mathematics

“ A great part of its higher arithmetic theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed.”

The simple methods become available when the right objects and the proper definitions have been understood at last. The profound theorems, those with important consequences, are proved then almost trivially. Fermat’s theorem may not be so profound in what is states as in what it touches on. In working to provide a proof many results with broad applicability were uncovered.

]]>