
Feynman on Fermat's Last Theorem (2016) - slbenfica
http://www.lbatalha.com/blog/feynman-on-fermats-last-theorem#email
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merricksb
Previous HN discussion around the time of publication just over a year ago:

[https://news.ycombinator.com/item?id=12018221](https://news.ycombinator.com/item?id=12018221)

~~~
dang
Also
[https://news.ycombinator.com/item?id=14355834](https://news.ycombinator.com/item?id=14355834).

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jordigh
Sigh, this proof bugs me so much.

Compute an approximate distance between nth powers, interpret this as the
probability of an integer being an nth power, integrate this probability over
the sum x^n + y^n, see that the probability of this being an nth power is also
very low.

I guess this is close enough for government work, but it's so utterly
fallacious. For example, the distance between n^2 and (n-1)^2 is 2n - 1. That
"means" that the "probability" of N being a perfect square is about 1/(sqrt(2N
- 1)). This probability also goes to zero in the limit as N goes to infinity.
Not very quickly, but it does.

Does that mean that square numbers don't exist?

We have many examples of conjectures being disproved by very large
counterexamples:

[https://www.quora.com/What-is-an-example-of-a-conjecture-
tha...](https://www.quora.com/What-is-an-example-of-a-conjecture-that-was-
proven-wrong-for-very-large-numbers)

~~~
pmiller2
I have to agree that this argument is utterly unconvincing, even as a reason
to believe FLT _may_ be true intuitively. It simply dismisses with a wink and
a nod the idea that there might be one single counterexample out there for one
single exponent (or, a finite number of counterexamples for a finite number of
exponents). In other words, it does nothing to argue against the set of
counterexamples being of measure 0. Indeed, until Wiles finally proved FLT,
the relative lack of progress on Siegel's conjecture on the infinitude of
regular primes would tend to lend support to the view that a counterexample
might exist somewhere out there.

~~~
alimw
I haven't read it closely, but it looks to me as if the calculation estimates
the expected _number_ of counterexamples rather than the "measure" of them
(however you've chosen to define that).

~~~
mathemancer
The set of possible counterexamples has Lebesgue measure zero.

[https://en.wikipedia.org/wiki/Lebesgue_measure](https://en.wikipedia.org/wiki/Lebesgue_measure)

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faitswulff
Feynman's approach seems very reminiscent of Jake Vanderplas's Statistics for
Hackers talk[0] as opposed to the purely theoretical physicist approach that
the author notes at the end.

[0]:
[https://www.youtube.com/watch?v=Iq9DzN6mvYA](https://www.youtube.com/watch?v=Iq9DzN6mvYA)

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throwanem
> the main job of theoretical physics is to prove yourself wrong as soon as
> possible.

The main job of practical engineering, too.

~~~
auggierose
Nah, I'd say the job of practical engineering is to provide reliable, correct
solutions in a timely, in-budget manner.

If you are talking about theoretical engineering, I agree :D

~~~
throwanem
> the job of practical engineering is to provide reliable, correct solutions
> in a timely, in-budget manner

Sure. And I'm sure you'll agree this job is easier to do in direct proportion
to the celerity with which we eliminate impractical approaches.

~~~
auggierose
You can eliminate impractical approaches all day long. At some point you have
to start with the practical approach that you know will work a-priori.

~~~
throwanem
When you have the option, sure! Not all problems are so tractable.

~~~
auggierose
Granted. Still, I think there is a fundamental difference: In theoretical
physics, a good theory of how things work (or how they definitely don't work)
is enough as the result of your work, you don't need a working product in the
end. In practical engineering, you do!

Let me also add that I am serious about theoretical engineering, actually
that's one of my favourite occupations, together with practical engineering.

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jwilk
Here's a more straight-forward (but equally hand-wavy) way to calculate the
probability that N is a perfect power:

P(N) ≈ (number of perfect powers near N) / (size of the neighborhood)

≈ (ⁿ√(N + r) − ⁿ√N) / r, for some smallish r

≈ d/dN (ⁿ√N)

= ⁿ√N / nN

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chmaynard
Apparently the author also hosts this interesting website:
[http://www.fermatslibrary.com](http://www.fermatslibrary.com)

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kumarvvr
Geniuses stand on the shoulders of giants and look further ahead, while I look
at giants and assume a fetal position and cower in fear. Sigh !

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Gregaros
>Richard Feynman was probably one of the most talented physicists of the 20th
century.

Starts off bold.

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joombaga
Do you disagree? Would it help for me to show you the probability is strictly
non-zero?

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Gregaros
Sarcasm. "Probably one of the most" is quite a few weasel words for somebody
universally included in any list of significant physicists.

