Linked Questions
17 questions linked to/from Awfully sophisticated proof for simple facts
215
votes
67
answers
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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 ...
63
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 ...
- 1,501
86
votes
6
answers
16k
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 ...
- 10.8k
49
votes
4
answers
8k
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 ...
- 43.5k
20
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 ...
- 3,397
65
votes
1
answer
6k
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 ...
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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 ...
- 159
15
votes
5
answers
2k
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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 $p$. ...
- 27k
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-...
- 14.8k
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?...
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4
votes
1
answer
765
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 ...
- 33.2k
8
votes
1
answer
507
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 $\mathcal{B}(F)$ ...
- 724
7
votes
1
answer
467
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 ...
- 181
3
votes
1
answer
268
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 $\mathbb{R}$ ...
- 44.5k
16
votes
0
answers
737
views
Is there a "natural" proof of the equality $4^2=2^4$?
This question, or rather any answer that it might receive, would probably belong to the realm of Awfully sophisticated proof for simple facts. Still, I claim that I have quite serious motivation for ...
- 17.3k
7
votes
0
answers
310
views
Proving infinitely many primes using algebraic geometry ideas
There are at least two well known proofs of the infinitude of primes (Euclid's original one and Euler's proof using L-series) and both of them can be extended to prove more general statements of the ...
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