# Questions tagged [alternative-proof]

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

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### First known proof of the $2 \cdot n-2$ Theorem for the planar generalization of the Nine dots problem

Reading the Wikipedia page about the well-know Nine dots puzzle, I have just seen that the planar generalization of this problem would have been proven in 1956 (see Wikipedia: Nine dots puzzle), while ...
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### Question regarding proof of Littlewood-Paley

I posted this question on Math.SE where I unfortunately received no answers even after a bounty. As such, I am putting it here, in hopes to receive a response. For the proof of Theorem 6.1.6 in ...
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### Proof of the Dunford-Pettis theorem in the context of probability spaces

I'd like to know if there's a proof of the Dunford-Pettis theorem without using relatively advanced theorems of functional analysis such as Eberlein–Smulian Theorem. Since I'm only interested in ...
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### Conceptual explanation for extra/missing $p$ solutions to $x^2+y^2=a \pmod p$ at $a=0$

Throughout, $p$ will denote a prime integer, and $k$ an arbitrary integer. I have worked through V. Lebesgue's proof of quadratic reciprocity outlined by Keith Conrad in this MO thread, and I feel ...
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### Companion matrices must have geometric multiplicity one, linear recurrence sequence view [closed]

I posted this question on math stackexchange weeks ago, and it have not receive an answer yet after a bounty offer... I've been recently playing around with the linear recurrence sequences. Consider ...
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### A simple proof of the fundamental theorem of Galois theory

Update. It's now on the arXiv. Some time ago I found my "own" proof of the fundamental theorem of Galois theory. You can find a pdf with the proof (link removed, see arXiv). It is quite ...
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### Examples of "proof by generalising" [duplicate]

In a previous post I asked (Which theorems have Pythagoras' Theorem as a special case?). Are there any compelling examples where it is significantly "easier"/"simpler" to prove ...
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### Alternative proofs of Euclid-Euler theorem

What are some alternative methods of proof for the necessity direction of the above theorem, ie $n$ an even perfect number $\Rightarrow n$ is of form $2^{a-1} (2^a - 1)$ where $2^a - 1$ is a Mersenne ...
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### Has Goedel's Second Incompleteness Theorem been proven using Lawvere's Fixed Point Theorem?

This question is a request for assistance in surveying the existing literature on applications of Lawvere's Fixed Point Theorem (LFPT). Yanofsky [0] has demonstrated several applications of LFPT to ...
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### Why can any open subset $U$ of $\mathbb{Q}^\infty$ be written as disjoint union of basic clopen subsets?

I am reading Engelen´s paper and have trouble with this proof of Lemma 2.1 (a) (link is below). It is easily seen that any non-empty open subspace $U$ of $\mathbb{Q}^\infty$ can be written as an ...
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### Which theorems have Pythagoras' Theorem as a special case?

Loomis famously wrote hundreds of proofs of Pythagoras' Theorem (reference below), but these are all basically proofs "from below". Today on Twitter @panlepan mentioned Carnot's theorem ...
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### Verification of a certain computation of VC dimension

Disclaimer: I'm not very familiar with the concept of VC dimensions and how to manipulate such objects. I'd be grateful if expects on the subject (learning theory, probability), could kindly proof ...
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Recall that a quadrisecant of a knot is a line that passes thru four points on it. If the points appear on the line in the order $a$, $b$, $c$, $d$ and on the knot in the order $a$, $c$, $b$, $d$, ...
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### Collecting alternative proofs for the oddity of Catalan

Consider the ubiquitous Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$. In this post, I am looking for your help in my attempt to collect alternative proofs of the following fact: $C_n$ is odd if and ...
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### Results with short, advanced proofs or long, elementary proofs

Recently I was preparing an undergrad-level proof of (a form of) the Jordan Curve Theorem, and I had forgotten just how much work is involved in it. The proof stored my head was just using Alexander ...
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### A question on the applicability Chebyshev inequality for sequence of random quantities

Let $(X_n)_n$ and $(Y_n)_n$ be two mutually independent sequences of random tensors (i.e scalars, vectors, matrices, etc.) defined on the same probability space, and let $f$ be a measurable function. ...
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### Which step is wrong in the following simplification of Silver's forcing?

Theorem: If M is a countable transitive model of ZFC, and $\kappa$ is a supercompact cardinal in M, and $2^\kappa=\kappa^+$. Then there exists a forcing extension M[G] such that $\kappa$ becomes a ...
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### An alternate definition of Sobolev space $W^{1,p}(\Omega)$ when $1<p\leq\infty$ and consequences

Suppose that we define the Sobolev space $W^{1,p}(\Omega)$ with $1<p\leq \infty$, where $\Omega\subset\mathbb{R}^d$ ($d\geq 1$) is an open set (not necessarily bounded), in the following manner. ...
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### 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 $X$” ...
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### 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(\...
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### 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 ...
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