# Questions tagged [alternative-proof]

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

53
questions

**13**

votes

**1**answer

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### Topological proof that a Vitali set is not Borel

This question is purely out of curiosity, and well outside my field — apologies if there is a trivial answer. Recall that a Vitali set is a subset $V$ of $[0,1]$ such that the restriction to $V$ of ...

**22**

votes

**4**answers

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### What's the use of countable ordinals? (prompted by a remark of Tim Gowers)

In a typically lucid and helpful page of notes for students, A beginner’s guide to countable ordinals, Tim Gowers explains how the countable ordinals can be “constructed rigorously in a way that ...

**-1**

votes

**1**answer

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### Any simplification of this inequality if it is true? :For $t\geq 1.22$: $|\zeta(0.5+it)|\leq 0.5 \frac{|\Gamma(0.5+it)|}{|\Gamma(-0.5+it)|}$

When I tried to give bounds for $\zeta(0.5+it)$ using some transformations over Gamma function using the function $f(x)=\exp(-n x)$ over the range $(0,+\infty)$ , For $ Re(s)=\frac12 $ and $t >0$ ...

**9**

votes

**0**answers

344 views

### Including alternative proofs

Suppose I have found two or even more proofs of a theorem and I prepare a paper on it. Is it considered to be a good practice to write down all of them? Or is it considered to be my job as an author ...

**3**

votes

**0**answers

277 views

### Understanding a part of Friedberg’s Priority Argument Paper

This is Richard Friedberg’s original 1957 proof of the Friedberg-Muchnik Theorem, the origin of the ground-breaking priority argument. (The result proven is that there are two recursively enumerable ...

**7**

votes

**3**answers

768 views

### Euler's rotation theorem revisited - Elementary geometric proofs

This is a very elementary topic but I thought it might be worth giving it a try here, I would be very interested in any comments - I originally posted it to Maths SE.
Euler's Rotation Theorem, proved ...

**2**

votes

**0**answers

158 views

### Proof of non-solvability of general equations of one unknown in elementary terms (finite terms)?

This question relates to the solvability of equations of one unknown in elementary terms (finite terms) according to Liouville and Ritt.
Elementary equations and closed-form solutions can be ...

**3**

votes

**0**answers

168 views

### A closed set which is the closure of its interior points

This is a cross-post to MSE to this question
https://math.stackexchange.com/questions/3267497/a-set-which-is-the-closure-of-its-interior-points
There is a related question in MO:
Closure of the ...

**6**

votes

**1**answer

261 views

### Severi Formula for Intersection Multiplicities

I say in advance that I am a novice in Intersection Theory, so forgive me if my question is trivial.
Let $X\subseteq\mathbb{P}^N$ be a smooth irreducible projective variety of dimension $n$ and $V, W\...

**4**

votes

**2**answers

581 views

### Hard implications that become easy with the right intermediate step

I’m interested in examples of theorems of the form “If $P$ then $Q$“ that were either unsolved or thought to require difficult arguments until someone came up with an $X$ for which “If $P$ then $X$” ...

**16**

votes

**5**answers

820 views

### Closed-form expression for certain product

$\mathrm G$ is Catalan's constant.
I recently found the product
$$
\alpha=\prod_{n=1}^{\infty}\frac{E_n(\frac12)E_n(\frac7{12})E_n(\frac1{20})E_n(\frac{13}{20})}{E_n(\frac14)E_n(\frac1{12})E_n(\...

**7**

votes

**1**answer

229 views

### Deuring's result on elliptic curves. Any proof reference

I have heard of this result from Deuring 1941 paper: Given $\mathbb F_p$ ($p$ prime number) and any number $n$ in the Hasse interval $[p+1-2\sqrt p, p+1+2\sqrt p]$ there is an elliptic curve over $\...

**4**

votes

**2**answers

265 views

### Orthogonal Polynomials and Sturm Liouville operators

Classical Orthogonal polynomials (e.g., Hermite, Legendre) are eigenfunctions of Sturm Liouville operators. For example, define $L[u]=u''-xu'$, then the $n$-th order Hermite polynomial satisfies $...

**0**

votes

**1**answer

246 views

### Fibers and Push-forward

Let $f\colon X\longrightarrow Y$ be a surjective morphism of projective varieties. Consider a non-singular irreducible projective curve $T$ and surjective morphism $p_Y\colon Y\longrightarrow T$. Then ...

**29**

votes

**2**answers

1k views

### Why did Dedekind claim that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ hadn't been proved before?

In a letter to Lipschitz (1876) Dedekind doubts that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ had been proved before:
quoted from Leo Corry, Modern algebra, German original:
Why did Dedekind doubt that $(\...

**3**

votes

**2**answers

285 views

### Alternate descriptions of finite fields

The finite field of order $p^n$ is isomorphic to $(\mathbb Z/p \mathbb Z)[X]/(P)$, where $P$ is an irreducible polynomial in $(\mathbb Z/p \mathbb Z)[X]$ of degree $n$. This describes every finite ...

**3**

votes

**2**answers

242 views

### A direct proof that every $r$-colored complete graph on $n=(r+1)m-(r-1)$ vertices has a monochromatic matching of size $m$?

Cockayne and Lorimer ("The Ramsey number for stripes" 1975) prove that in every $r$-colored complete graph on $n=\sum_{i=1}^rm_i+m_1-(r-1)$ vertices, where $m_1\geq \dots\geq m_r\geq 1$, has a ...

**7**

votes

**1**answer

242 views

### If $M \otimes -$ is continuous, why is $M$ f.g. projective? Alternative proof

Let $R$ be a commutative ring and $M$ be some $R$-module such that $M \otimes -$ is continuous (i.e. preserves all limits). Then one can show that $M$ is f.g. projective.
One way to prove this is to ...

**11**

votes

**1**answer

340 views

### What was Smith's proof of Smith's theorem on Hamilton cycles in cubic graphs?

In a short 1946 paper "On Hamiltonian Circuits", Tutte proved the famous result that an edge in a cubic graph lies in an even number of Hamilton circuits.
He attributed the result to his friend CAB ...

**0**

votes

**0**answers

129 views

### Simple proof for Riemann/Hurwitz ζ functional equation

For the purpose of formalisation, I am looking for a simple proof of the function equation of the Riemann ζ function or the generalisation thereof, the Hurwitz's formula for the Hurwitz ζ function.
...

**0**

votes

**1**answer

207 views

### How can i justify this multinomial coefficient identity? [closed]

$$\sum_{ k\ge 1 } \sum_{ (s_1,...,s_k) } \binom h{ s_1 ,..., s_k }=\binom{m-1}{h-1}$$
where the second summation is taken over all choices of the numbers
$${ s }_{ 1 },...,{ s }_{ k-1 }\ge 0,\quad {s ...

**52**

votes

**12**answers

4k views

### Examples of advance via good definitions

In my research I came across a case where I could derive a known theorem with rather straightforward way by choosing "non-standard" definitions using my knowledge from a related field. This particular ...

**24**

votes

**3**answers

4k views

### Where to publish a new proof of an old theorem?

A few months ago I came up with a proof for an old theorem. After being excited for a moment, I then tried to find my proof in the literature. Since I did not find it, then I started to wonder if it ...

**11**

votes

**1**answer

526 views

### On Selberg's Proof of the Selberg Integral Formula

I'm attempting to read through Mehta's write-up of Selberg's proof of the formula for the Selberg integral formula given below:
$$
\begin{align}
\operatorname{S}_n(\alpha,\ \beta,\ \gamma) & = \...

**3**

votes

**0**answers

107 views

### Colimits of algebras of an endofunctor

I try to understand a proof in Adamek-Rosicky's book "Locally presentable and accessible categories", Cambridge University Press 1994. In Corollary 2.75 (p. 121) it is proven that the category $\...

**-2**

votes

**1**answer

239 views

### Is there a proof (maybe formulated by Feferman) which says that a proof about the (in)consistency of ZFC is unachievable? [closed]

Is there a proof (maybe formulated by Feferman) which says that a proof about the (in)consistency of ZFC is unachievable?
A professor said it to me a long time ago, but I don't have any references.
...

**8**

votes

**2**answers

754 views

### Link between the hairy ball theorem and the fundamental theorem of algebra

I read in the book "Concepts of modern mathematics" by Ian Stewart that it was possible to proof the fundamental theorem of algebra using the hairy ball theorem (complete reference to the page is in ...

**19**

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**3**answers

1k views

### Simplicity of alternating group $A_n$

I am teaching an introductory group theory course, and it has come to the inevitable proof that $A_n$ is simple for $n\geq 5$. Now, there seem to be a number of proofs that I can find – one the "...

**21**

votes

**2**answers

803 views

### Evaluating an integral using real methods

This is a bit of recreational integration. The following, rather attractive integral is quite straightforward via residues:
$$\int_0^1 x^{-x}(1-x)^{x-1}\sin \pi x\,\mathrm{d}x=\frac{\pi}{e}$$
...

**1**

vote

**0**answers

126 views

### Another proof of the comparison principle for PDEs by the theory of viscosity solutions

In my research I study (fully) nonlinear PDEs with fractional derivatives. For an analysis of these, I use the theory of viscosity solutions.
So I am now at the end of my tether becasuse I can not ...

**4**

votes

**0**answers

141 views

### Undecidability of the existential theory

Do you know if I can find the proof that the existential theory of $\mathbb{Z}$ with the structure of addition , divisibility and the relation $(\exists s \in \mathbb{Z})m=np^s$ is undecidable, ...

**11**

votes

**2**answers

1k views

### Is there a fixed integer $x>1$ satisfing ${\sigma}^{k}(x)\equiv 0\pmod{x}$ for all positive integers $k$?

This question related to this question from SE. I'm interested to know if there exists an integer $x>1$ that satisfies $${\sigma}^{k}(x)\equiv 0\pmod{x}$$
for all positive integers $k$.
Note. $\...

**8**

votes

**1**answer

344 views

### Does the following $ C^{*} $-algebraic result have a purely algebraic proof?

While studying the proof of Bott periodicity for operator $ K $-theory in this set of notes, I learned this fact:
Theorem. Let $ A $ and $ B $ be $ C^{*} $-algebras. Let $ f,g: A \to B $ be $ * $-...

**4**

votes

**1**answer

277 views

### Subfields of $\mathbb{Q}\bigl(\sqrt[n]{a}\bigr)$ for $a>0$

This is related to a question on Math Stack Exchange.
Given a rational number $a>0$ and an $n\in\mathbb{N}$ such that $x^n - a$ is irreducible over $\mathbb{Q}$, it is known that every subfield of ...

**8**

votes

**3**answers

705 views

### How to prove that two univariate polynomials are always algebraically dependent?

How to prove that two univariate polynomials(over any field) are always algebraically dependent? Also, how to prove the generalization of this question i.e if number of polynomials are more than ...

**26**

votes

**7**answers

4k views

### Conceptual algebraic proof that Grassmannian is closed in Plucker embedding

I'm planning lectures for my intro algebraic geometry course, and I noted something awkward that is coming up. We're starting projective varieties soon. Of course, we'll prove that projective maps are ...

**2**

votes

**0**answers

134 views

### seek another proof of a result in Fourier analysis

It was proved on page 26 of this note the following result:
Let $\xi$ be an algebraic number that is not a root of unity, then there exists an $n_0\geq 0$ with the property that $$\beta=\sum_{j=-n^2}^...

**14**

votes

**1**answer

716 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 ...

**6**

votes

**2**answers

299 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

**2**answers

320 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

**1**answer

252 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 ...

**22**

votes

**12**answers

2k 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. ...

**12**

votes

**1**answer

1k 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
...

**33**

votes

**7**answers

9k 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 ...

**24**

votes

**3**answers

3k 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.

**1**

vote

**1**answer

2k 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

**1**answer

265 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} \frac{\text{Vol}(B^n(r))}{\text{Vol}...

**9**

votes

**1**answer

2k 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 ...

**27**

votes

**9**answers

14k views

### Direct proof of irrationality?

There are plenty of simple proofs out there that $\sqrt{2}$ is irrational. But does there exist a proof which is not a proof by contradiction? I.e. which is not of the form:
Suppose $a/b=\sqrt{2}$ ...

**107**

votes

**36**answers

26k views

### Quick proofs of hard theorems

Mathematics is rife with the fruit of abstraction. Many problems which first are solved via "direct" methods (long and difficult calculations, tricky estimates, and gritty technical theorems) later ...