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Almost Sure
9.1K posts
Almost Sure
@Almost_Sure
George Lowther, Author of Almost Sure blog, on maths, probability and stochastic calculus. Also on YouTube https://t.co/VyOijwbe9l
London, UK Katılım Eylül 2020
251 Takip Edilen8.8K Takipçiler
the genus is (n-1)(n-2)/2 so has genus > 1 only for n ≥ 4.
The case n=3 is different. FLT still applies but, algebraically, it’s a different kettle of fish. An elliptic curve, which could potentially have infinitely many solutions, but here the rank is 0, so only finitely many.
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An example application of Faltings’ theorem is that for each integer n ≥ 4
xⁿ + yⁿ = 1
has only finitely many rational solutions. That’s because it’s a curve of genus > 1.
ofc, Fermat’s Last Theorem says only solutions are x=0 or y=0, but Wiles’ proof is relatively recent
Almost Sure@Almost_Sure
Congrats to Gerd Faltings! And if you want to know how to prove Faltings’ theorem (formerly Mordell conjecture), check out this …ahem… “introduction” to the theory!
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@jamestanton Reducing to lowest terms, a² and b² must be in {1,4,7} modulo 9. No pair of these sum to 0 mod 9. So, no
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A power of three can be the side-length of an integer right triangle. (Why?)
Can it be the hypotenuse of one?
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@Almost_Sure Hindry and Silverman however give a different proof (Bombieri's version of Vojta's, using Diophantine approximation), not the original one by Faltings. For the latter, see the 1986 volume edited by Cornell and Silverman that includes an English translation of Falting's original.
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Congrats to Gerd Faltings!
And if you want to know how to prove Faltings’ theorem (formerly Mordell conjecture), check out this …ahem… “introduction” to the theory!
The Abel Prize@abel_prize
We congratulate Gerd Faltings as the 2026 Abel Prize laureate! 🎉 He recives the Abel Prize "for introducing powerful tools in arithmetic geometry and resolving long-standing diophantine conjectures of Mordell and Lang".
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and visualizing a cubic curve
(actually, I’ll port this to algan, as manim doesn’t do proper hidden surface removal)
Almost Sure@Almost_Sure
Using Claude to generate some animations… looking pretty good. Not perfect, there seems to be some artefacts which may be down to FFT use.
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Apparently the British make charcuterie boards but with leftovers and call them "picky bits" it's very endearing.
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There is no fucking way you’d ever guess whose Wikipedia page this is from.
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Fixing aliasing issue and increasing resolution. Except for some text overlapping, it looks good!
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Asking it to use exact formulas instead of FFT. A little better but still doesn’t look quite right, will need manually checking
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Using Claude to generate some animations…
looking pretty good. Not perfect, there seems to be some artefacts which may be down to FFT use.
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Well, X is the time that Poisson process of rate 1/b hits value n, and Y is its value at time x
Coupling!
Probability Fact@ProbFact
If X ~ gamma(n, b) and n is an integer, then P(X <= x) = P(Y >= n) where Y ~ Poisson(x/b).
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@jamestanton only possibilities are multiples of
(2ⁿ⁺¹)² + (4ⁿ-1)² =(4ⁿ+1)²
for n=1,…99.
So answer is 99
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How many integer right triangles have a side of length 2^100?
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@BrunsJulian1541 ok, but it remains, that if they can’t solve it properly for whatever reason, they might just make up a wrong answer and then claim a proof of the wrong answer
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@Almost_Sure all of that is contained in their training data, no online search required.
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From the responses it seems that the higher tiers of chatbots correctly solve this.
However, I’m suspicious that if they couldn’t find sufficient info online, they might still just guess an answer and claim it’s correct
Almost Sure@Almost_Sure
So ChatGPT gets the wrong answer, then *proves* the wrong answer. Then when a counterexample is pointed out, calls it *my* assumption that was wrong 🤷 do the more advanced versions that you subscribe to handle this better?
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