#fermatslasttheorem — Public Fediverse posts

Andrew Wiles on proving Fermat’s Last Theorem (1995) [video]

https://www.youtube.com/watch?v=GS7CxAtV5Ks

Comments: https://news.ycombinator.com/item?id=49203626

#HackerNews #AndrewWiles #FermatsLastTheorem #math #video #theorem #proof

Results ranging from visualizable theorems of solid geometry to abstract propositions of analysis were called beautiful by Leonhard Euler (1707–83). For instance, he thought beautiful the following result:

If an elliptical cylinder is cut by any plane at an angle θ, then the ratio of the product of the principal axes of the section and of the product of the principal axes of the base is 1:cos θ (see attached image).

Aesthetic concerns seem to have been part of what drew Euler to number theory. Christian Goldbach (1690–1764) persuaded him to take an interest in the subject and to make a serious study of Fermat's work. His attention was drawn by the theorem:

Every natural number can be expressed as a sum of four squares.

With presumably deliberate understatement, Euler described it as a ‘not inelegant theorem’. The result remained unproven in Euler's time, and the first proof was given by Joseph-Louis Lagrange (1736–1813), becoming known as ‘Lagrange’s four-square theorem’.

Thus, for Euler, *unproven* conjectures could have aesthetic value. And so he judged another well-known then-unproven result of Fermat:

‘In Fermat there is another very beautiful theorem for which he claims to have found a proof. […] the formula $a^n + b^n = c^n$ is impossible whenever $n > 2$’

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[Each day of February, I am posting a short interesting story/image/fact/anecdote related to the aesthetics of mathematics.]

#Euler #Fermat #Goldbach #Lagrange #FermatsLastTheorem #MathematicalBeauty

Hi fam, have a fulfilling day! Here comes #Pythagoras 2.0 and a proof of #Fermat'sLastTheorem. Finding every #PythagoreanTriplet that includes any #naturalNumber >2 has never been easier! (0.5d(m²-1))²+d²m²=(a+d)² with 0.5d(m²-1) as "a", dm as "b", 0.5d(m²+1) as "c" #education #mathematics #math

"And then the first sentence of chapter 1 of the paper proper is "Consider quasi-finite separated commutative group schemes of finite presentation over the base \(S:= \mathrm{Spec}(\mathbb{Z}) \) which are finite flat group schemes over \(S':= \mathrm{Spec}(\mathbb{Z}[1/N]) \)". At the time of writing (May 2024), Lean’s algebraic geometry cannot get us through the first sentence of Mazur’s proof, which occupies pages 43 to 172 of the paper (not including the appendix or references, that’s just the proof). Anyone interested in formalising Mazur’s paper should make a formalisation of its first sentence their first milestone."
— @xenaproject

#FermatsLastTheorem