Looking for a proof different from the standard proof(s) of a result

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13
votes
1answer
350 views

Gabber's proof of Br' = Br for quasiprojective schemes

In a note by deJong showing the cohomological and ordinary Brauer groups coincide for separated quasicompact schemes with ample line bundle, it is mentioned that Gabber had an unpublished proof of the ...
5
votes
2answers
149 views

A few standard results (on metrizability and relative separation strength) without Choice?

I've been going back over some results from Munkres's Topology, and I'm curious about some things. (I originally posted this on M.SE, but I think it is probably a better fit here.) I know that Choice ...
2
votes
2answers
240 views

Stronger theorem not resulting from proof analysis

Suppose that we proved $\varphi$ from a theory $T$. Often we ask whether or not we could have proved $\varphi$ with a weaker theory, to find out we usually analyze the proof and try to figure out ...
4
votes
1answer
233 views

Alternative proof for counting problem in graphs

Let $G$ and $H$ be graphs, let $\vec H$ be a fixed orientation of $H$. Denote by $D(G,\vec H)$ the number of orientations of $G$ that contain a copy of $\vec H$ and denote by $D'(G,H)$ the number of ...
15
votes
12answers
1k views

Instances where an existence result precedes the constructive version

The basic motivation here is to encourage and inspire - via examples - the pursuit of alternate proofs of existing results that might be more accessible and intuitive by cataloging success stories. ...
9
votes
1answer
984 views

What is the protocol for making modifications to someone else's proof to prove something slightly stronger?

I have a need to modify Erdős' proof of the Sylvester-Schur Theorem to prove something stronger. See my working document at http://math.rudytoody.us/ or http://math.rudytoody.us/OppermannTheorem.pdf ...
21
votes
6answers
4k views

What are some proofs of Godel's Theorem which are *essentially different* from the original proof?

I am looking for examples of proofs of Godel's (First) Incompleteness Theorem which are essentially different from (Rosser's improvement of) Godel's original proof. This is partly inspired by ...
18
votes
2answers
1k views

Riemann mapping theorem for homeomorphisms

How do you prove to any two simply-connected domains in the plane are homeomorphic without using the Riemann mapping theorem? An elementary proof would be nice.
0
votes
1answer
933 views

The Hexagonal Property of Pascal's Triangle

Any hexagon in Pascal's triangle, whose vertices are 6 binomial coefficients surrounding any entry, has the property that: the product of non-adjacent vertices is constant. the greatest common ...
1
vote
1answer
229 views

Synthetic Proof for Ratio of Volumes of Concentric Spheres?

Let $B^n(r)$ be the $n$-ball of radius $r$. A standard (easy) problem for first year calculus students is the following. $(1)$ Show that $$ \lim_{n\to \infty} ...
4
votes
1answer
1k views

Non-trivial consequences of Baer's theorem and Lucchini's theorem in subnormality theory

There are a couple of beautiful results in finite group theory that look trivial, at least on a first glance, but require non-trivial facts to prove. I am basically interested in whether these results ...
15
votes
3answers
2k views

Different way to view action of fundamental group on higher homotopy groups

There are a couple of ways to define an action of $\pi_1(X)$ on $\pi_n(X)$. When $n = 1$, there is the natural action via conjugation of loops. However, the picture seems to blur a bit when looking at ...
5
votes
5answers
1k views

Alternative proof of unique factorization for ideals in a Dedekind ring

I'm writing some commutative algebra notes, but I'm facing a difficulty in organizing the order of the topics. I'd like to have the topics about factorization before speaking of integral closure. This ...
14
votes
3answers
3k views

Quick proof of the fact that the ring of integers of Q(\zeta_n) is Z[\zeta_n]?

I cannot find a good reference for the proof that the ring of integers in a cyclotomic field $\mathbb{Q}(\zeta_n)$ is $\mathbb{Z}[\zeta_n]$. The proof I usually find does an induction on the number of ...