#euler — Public Fediverse posts

Achievements in maths

https://piefed.social/c/historymemes/p/2027504/achievements-in-maths

E, já que estamos falando tanto em #Euler hoje, nada melhor do que corrigir trabalhos tomando #cappuccino na minha xícara do Euler rs.

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 $n > 2$’

1/2

[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

Zum #Euler-Tag ein kleines #Calliopemini-Projekt gebaut (im Simulator von MakeCode nutzbar):
https://makecode.calliope.cc/_YvdYr29wmdWh

<Hav_e_fun/>!

// CC @calliope

It’s incredible that these two identities can explain so much.

Link: https://arxiv.org/abs/2505.03754

#math #calculus #euler #unification #generalization #integrals #integration

Weierstrass Substitution as a Special Case of the USM Framework:
https://geometriadominicana.blogspot.com/2025/12/weierstrass-as-special-case-of-usm.html

#math #calculus #Weierstrass #USM #framework #euler #unification #integral #integration

Никлаус Вирт и язык Pascal — легенды 80-х

В 1970 году молодой швейцарский учёный и программист Никлаус Вирт (Niklaus Wirth) выпустил первую версию Pascal. Прошло более полувека, автор умер в 89 лет, а вот Паскаль остаётся актуальным и популярным языком программирования.

https://habr.com/ru/companies/ruvds/articles/968706/

#Никлаус_Вирт #Pascal #Turbo_Pascal #Delphi #Niklaus_Wirth #Free_Pascal #Lazarus_IDE #Algol #Euler #ruvds_статьи

Of course this was known to many famous #mathematicians, like #Euler and #Jacobi and they solved the equations in terms of #EllipticIntegrals. I find it satisfying to be able to visualize the #dynamics of the this system, using #FreeSoftware, such as #WxMaxima.

USM really does slash the rote burden, chiefly because one handful of exponential/hyperbolic identities replaces a patchwork of separate trig, inverse-trig, radical and Euler recipes.

Article (draft): https://arxiv.org/abs/2505.03754

#math #calculus #integral #new #euler #arxiv #halfangleapproach #symmetrymatters

One day, one decomposition
A228183: Semiprimes generated by the Euler polynomial x^2 + x + 41

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A228183.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A228183.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #numbers #semiprimes #Euler #polynomial #graph #threejs #webGL

The USM, the Dominican method 🇩🇴, has relegated Euler substitutions to mere historical relics. Modern integration has a new name.

Method draft: https://drive.google.com/file/d/12DayP6cD1VwDIZCL-nMlcaNH2XUwHfAy/view?usp=drivesdk
#math #calculus #integrals #method #euler #halfangleapproach #symmetrymatters

Euler–Mascheroni constant! :euler:

In fact, the last one is:
\[\large\displaystyle\int_1^{+\infty}\mathrm dx\ \left(\frac{1}{\lfloor x\rfloor}-\frac1x\right)=\gamma\approx0.5772156649\]

Equivalently,
\[\large\displaystyle\lim_{n\to\infty}\left(\sum_{k=1}^n \frac1{k}-\ln n\right)=\gamma=0.5772156649\ldots\]
---------------------------------------------------------------------------------------------------------------------
Unsolved problem in mathematics:
Is Euler–Mascheroni constant irrational? If so, is it transcendental?

#Euler #Mascheroni #EulerMascheroni #Constant #gamma #EulerConstant #EulersConstant #EulerMascheroniConstant #Irrational #Irrationality #Transcendental #Transcendence #Unsolved #UnsolvedProblem #Maths #Mathematics #Indeterminate #IndeterminateForm #IndeterminateForms #Inf #Infinity #HarmonicNumber #HarmonicNumbers #HarmonicSeries #Logarithm #Log #Logarithms #NaturalLogarithm #Integral #ImproperIntegral

One day, one decomposition
A145292: Composite numbers generated by the Euler polynomial x^2 + x + 41

3D graph, threejs - webGL ➡️ https://decompwlj.com/3Dgraph/A145292.html
2D graph, first 500 terms ➡️ https://decompwlj.com/2Dgraph500terms/A145292.html

#decompwlj #math #mathematics #sequence #OEIS #javascript #php #3D #composite #numbers #generated #Euler #polynomial #graph #threejs #webGL

#GregoryChaitin - Is #Mathematics #Eternal?

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

#Philosophy #PhilosophyOfScience #PhilosophyOfMathematics #Science #Math #Maths #Cantor #Ramanujan #SetTheory #God #Spirituality #Beauty #Transcendence #Numbers #Contradiction #Logic #Poincaré #Poincare #Euler #CloserToTruth #RobertKuhn

We made our first try to #simulate an #Euler #fluid within #NumeRe. Runs pretty well and displays also the expected Kármán vortex street.

What do you think? Learn more at https://en.numere.org

Euler Finance's offer to hacker: Keep $20M or face the law - The hacker committed a $196 million flash loan attack on the Ethe... - https://cointelegraph.com/news/euler-finance-s-offer-to-hacker-keep-20m-or-face-the-law #smartcontractaudit #ethereumcommunity #on-chainmessage #vulnerability #chainalysis #eulerlabs #flashloan #exploit #attack #threat #euler #code #trm #bug

> One of the central problems in fluid dynamics is to figure out if the [Euler] equations ever fail, outputting nonsensical values that render them unable to predict a fluid’s future states.
> Mathematicians have long suspected that there exist initial conditions that cause the equations to break down. But they haven’t been able to prove it.
> In a preprint posted online last month, a pair of mathematicians has shown that a particular version of the Euler equations does indeed sometimes fail.
...
> Perhaps in some situations, the equations will proceed as expected, producing precise values for the state of the fluid at any given moment, only for one of those values to suddenly skyrocket to infinity. At that point, the Euler equations are said to give rise to a “singularity” — or, more dramatically, to “blow up.”
> Once they hit that singularity, the equations will no longer be able to compute the fluid’s flow.
...

Computer Proof ‘Blows Up’ Centuries-Old Fluid Equations
https://www.quantamagazine.org/computer-helps-prove-long-sought-fluid-equation-singularity-20221116/

Very important and interesting work, if you're a mathematician, a physicist or an engineer that has ever 'crashed' with Euler/Navier-Stokes equations.

#math #physics #fluids #euler #navierstokes #equations #singularities