Some theorems

NON-THEOREM ANNOUNCEMENT: As someone struggling with depression/isolation, one reason I opened this account is to get some sense of belonging with the math community. I would appreciate if you can refrain from "that is obvious duh", "you miss the point" type of comments. Thanks.

[Bosboom–Demaine–Hesterberg–Lynch–Waingarten 2015]. It is PSPACE-hard to decide whether Player 1 has a forced win in a two-player Mario Kart race from given starting positions for the players.

Every position of Rubik's cube can be solved in at most 20 half-turn moves. Moreover there are positions for which 19 moves is not enough.

It is conjectured that all zeros are simple.

[Bui–Heath-Brown 2013]. If RH holds, then at least 19/27 of the zeros of the Riemann zeta-function are simple.

The upper bound is improved to 46 here: arxiv.org/abs/2409.15709

[Davis 1955]. Given a lattice (L, ≤), TFAE: (1) Every monotone f:L→L has a fixed point. (2) L is complete.

The product rule follows from the multivariate chain rule.

The unique pair consists of the right triangle with sides of lengths (377, 135, 352) and the isosceles triangle with sides of lengths (366, 366, 132).

A triangle is called rational if it has rational side lengths. [Hirakawa–Matsumura 2019]. Up to similitude, there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area.

@PeepNSheep1 Can you share it/its idea?

Update: it was, in fact, incredibly overkill and stupid. But it works! So I don't care

[Anderson 1977]. For a commutative ring R, every Artinian R-module is countably generated.

[Elkies–Klagsbrun 2024]. The elliptic curve y² + xy = x³ – 27006183241630922218434652145297453784768054621836357954737385x + 55258058551342376475736699591118191821521067032535079608372404779149413277716173425636721497 has rank ≥ 29.

@ModalMetamodel Source seems to be virtualmath1.stanford.edu/~conrad/modsem…

Algebraic number theorists should honestly be grateful that ultraproducts show up in arithmetic geometry. This is one instance. Roughly, we have to take a sort of ‘limit’ when no transition maps exist, like for a sequence of modular curves of levels that don’t divide each other.

[Galvin 1995]. Let E be an infinite set. Given a subgroup H of Sym(E), TFAE: (1) H is countable. (2) H ≤ ⟨f,g⟩ for some f,g∈Sym(E).

[Cohen–Moore–Neisendorfer]: For every odd prime p, the least power of p that annihilates the p-primary part of all the homotopy groups of S²ⁿ⁺¹ is pⁿ.

The ℤ-rank of the homotopy groups of a simply-connected finite CW-complex of dimension d either become zero in degrees ≥ 2d, or they exhibit exponential growth.

[Basu–Basu 2015]: The homotopy groups of a simply-connected closed topological 4-manifold are completely determined by its second Betti number.

@KristapsBalodi3 Artin–Wedderburn, Hopkins–Levitzki, Sylow existence

I'm planning on making a tier list video where I rank theorems, but only those suggested by you as a reply to this message, and suggestions on my youtube page. So please comment below what theorems you want to see!