Linked Questions

218 votes
67 answers
47k views

Proofs that require fundamentally new ways of thinking

I do not know exactly how to characterize the class of proofs that interests me, so let me give some examples and say why I would be interested in more. Perhaps what the examples have in common is ...
65 votes
4 answers
15k views

Proof that pi is transcendental that doesn't use the infinitude of primes

I just taught the classical impossible constructions for the first time, and in finding my class a reference for the transcendence of pi, I found a dearth of distinct proofs. In particular, those ...
Barry's user avatar
  • 1,521
86 votes
6 answers
17k views

What are the most elegant proofs that you have learned from MO?

One of the things that MO does best is provide clear, concise answers to specific mathematical questions. I have picked up ideas from areas of mathematics I normally wouldn't touch, simply because ...
59 votes
4 answers
7k views

Has Fermat's Last Theorem per se been used?

There is a long tradition of mathematicians remarking that FLT in itself is a rather isolated claim, attractive only because of its simplicity. And people often note a great thing about current ...
Colin McLarty's user avatar
50 votes
4 answers
9k views

Is there an "elementary" proof of the infinitude of completely split primes?

Let $K$ be a Galois extension of the rationals with degree $n$. The Chebotarev Density Theorem guarantees that the rational primes that split completely in $K$ have density $1/n$ and thus there are ...
François G. Dorais's user avatar
23 votes
10 answers
5k views

Completeness vs Compactness in logic

One standard approach to showing compactness of first-order logic is to show completeness, of which compactness is an easy corollary. I am told this approach is deprecated nowadays, as Compactness is ...
David Harris's user avatar
  • 3,475
72 votes
3 answers
8k views

Can you solve the listed smallest open Diophantine equations?

In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size is substitute 2 instead of all variables, absolute values instead of all coefficients, and ...
Bogdan Grechuk's user avatar
5 votes
1 answer
3k views

Has Stirling’s Formula ever been applied, with interesting consequence, to Wilson’s Theorem?

Pressing the envelope, presumably the best scenario would be a simple proof of the Prime Number Theorem. After all, Wilson’s Theorem gives a necessary and sufficient condition, in terms of the Gamma ...
Mike Jones's user avatar
16 votes
5 answers
2k views

Reference request: Recovering a Riemannian metric from the distance function

Let $M = (M, g)$ be a Riemannian manifold, and let $p \in M$. Writing $d$ for the geodesic distance in $M$, there is a function $$ d(-, p)^2 : M \to \mathbb{R}. $$ This function is smooth near $p$. ...
Tom Leinster's user avatar
  • 27.7k
5 votes
4 answers
4k views

Non-separable Banach space

The vector space $C_b(\mathbb R)$ of bounded continuous functions on $\mathbb R$ is non-separable: it is possible to produce a direct proof of this fact, mimicking the standard proof for the non-...
Bazin's user avatar
  • 16.2k
5 votes
3 answers
2k views

When is the graph of a function a dense set?

Let $f: \mathbb R \to \mathbb R$ be any function. When is the graph of $f$ dense in $\mathbb R^2$? The only examples I know for this are for non-measurable functions, but is that a necessary condition?...
Anindya's user avatar
  • 53
4 votes
1 answer
782 views

Use of infinitude of primes in the Green-Tao theorem [closed]

In a video I watched last night on nuking mathematical mosquitos, Matt Parker gave the following proof of the infinitude of primes: suppose there are finitely many primes. The Green-Tao theorem says ...
David Roberts's user avatar
  • 35.4k
8 votes
1 answer
522 views

Are the following subsets of a Hilbert space always homeomorphic?

Let $F$ be a infinite-dimensional complex Hilbert space, with inner product $\langle\cdot\;| \;\cdot\rangle$, the norm $\|\cdot\|$, the 1-sphere $S(0,1)=\{x\in F;\;\|x\|=1\}$ and let $\mathcal{B}(F)$ ...
Schüler's user avatar
  • 724
7 votes
1 answer
503 views

Combinatorial consequences of de Branges's Theorem?

I'm usually not a proponent of the mentality “here is a tool, what results can we prove with it?” (as I prefer to start at the other end with a well-motivated problem), but this famously entertaining ...
Erik Lundberg's user avatar
3 votes
1 answer
298 views

Extending a continuous self-map of an open subset to the whole space

We say that a $T_2$-space has the open extension property (OEP) if for any open set $U$ and continuous map $f:U\to U$ there is a continous map $g:X\to X$ such that $g|_U = f$. The space $\mathbb{R}$ ...
Dominic van der Zypen's user avatar

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