Questions tagged [alternative-proof]

If you already have a proof for some result but want to ask for a different proof (using different methods).

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Closed form for $\sum_{k=1}^\infty\frac{H_k^{(m)}}{k^n}$

Let's define $$\sigma(m,n)=\sum_{k=1}^\infty\frac{H_k^{(m)}}{k^n}$$ where $H_k^{(m)}=\sum_{n=1}^{k}\frac{1}{n^m}$ is the k-th generalized harmonic number of order $m$. In mathworld site Eq (20), I ...
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What are some elementary ways to show that there's no solution to $\pm1 \pm 2 \pm 3\dots \pm n =0$ for $n \equiv 1,2 \ \, (\bmod 4\,)$?

I was looking at sequence A063865 of the OEIS, which is given by the number of ways of choosing $+$ and $-$ signs such that $\pm1 \pm 2 \pm 3\dots \pm n =0$. If we define the $n$-th element of this ...
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Is this alternative proof that $\sqrt 2$ is irrational sound?

I have just taught the classic proof by contradiction that $\sqrt 2$ is irrational, and one of my students came up with the following proof: Assume that $\sqrt 2$ is rational, so $\sqrt 2=\frac{a}{b}$ ...
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Is there a simplified 4-color proof for pixilated maps?

In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same ...
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Proof about concave decreasing function with some constraints

Hi it's a follow up of How to prove that the functional equation $f(x)+f(y)=f(\frac{xf'(x)+yf'(y)}{f'(x)+f'(y)})+f(\frac{x+y}2)$ is verified only by some basic functions? : Conjecture :...
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$f$ is a polynomial and $f(0)\neq0$. Why is $fg+x$ a perfect square for some $g$?

On Hacker News, someone posted the following exercise: prove that every nonsingular complex symmetric matrix $M$ has a symmetric matrix square root. This is old chestnut. As the poster indicated, ...
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is proof by contradiction essential for proving the Pigeonhole Principle? [duplicate]

The standard way to prove the Pigeonhole Principle that, putting $n+1$ pigeons into $n$ pigeonholes necessitates the existence of at least one pigeonhole with at least 2 pigeons is by contradiction. ...
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Does the question that every group of order 77 must have an element of order 7 and an element of order 11 have an elementary proof

Question 21 of Chapter 10 of Abstract Algebra by Dan Saracino is as follows: Prove that every group of order 77 must have an element of order 7 and an element of order 11. The more general case of ...
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Exercise 4, Section 32 of Munkres’ Topology

Show that every regular Lindelöf space is normal. My attempt: Let $A$,$B$ be closed in $X$ such that $A \cap B=\emptyset$. By chapter 7 theorem 2.2 of Dugundji topology (equivalent definition of $T_3$...
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Examples of Intuitionistic Proof

For uni I need to write a paper and give a presentation on intuitionism, and I am looking for nice examples of theorems or other results to prove or disprove intuitionistically. One example I found ...
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Is there a shorter or more trivial way to prove that $x > \cos (x)-\cos (2 x)$ holds for all $x>0$?

I want to prove that the inequality $$x > \cos (x)-\cos (2 x)$$ holds for all $x>0$. My attempt: Since the function on the RHS is periodic, we can find the position of extrema (on first ...
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Possible ways of solving an Olympiad question

"A piece of land of a square shape with dimensions 10m x 10m is divided into 100 square parcels with dimensions 1m x 1m. Initially, 9 of the parcels are overgrown by weed. If a parcel is ...
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There exists a ball inside the convex hull of the union of $2$ other balls

I'm trying to show that given a point in the line segment connecting centers of $2$ given balls, there is a ball centered at that point inside the convex hull of the union of that $2$ other balls. ...
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Help me rewrite Theorem 3 from Serge Lang's Basic Mathematics using Velleman's 'given–goal diagrams'

I want to apply the proof strategies and overall scratch work diagram based framework introduced in the book "How to Prove It" in order to rewrite the following theorem: Theorem 3. Any ...
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A square on the equator of a sphere is a critical point of the electrostatic potential

$\newcommand{\S}{\mathbb{S}^2}$ This is a self-answered question. I learned something from spelling out the details, and I hope this could be interesting to others. I would welcome alternative ...
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The intuition behind this explicit form of an affine hull?

I have come across an explicit formula of an affine hull from here. I'm trying to prove that formula in 2. below. However, my approach is quite muscular and not natural, i.e. I have to know the ...
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Simpler proof that $y^3[d^2y/dx^2]$ is a constant if $y^2=ax^2+bx+c$?

here's my question If $y^2=ax^2+bx+c$ then prove that $y^3[d^2y/dx^2]$ is a constant . I have solved this using the conventional method, taking square root, differentiating w.r.t to x and using ...
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Question on analytic continuation of the principal branch of logarithms on $B(1, 1)$

(Excercise $6$ ,Conway ,page $217$ ) Let $D_0=B(1, 1)$ and $f_0$ be the restriction to $D_0$ of the principal branch of logarithm. For $n\in \Bbb{Z}$ let $\gamma(t) =e^{2\pi i nt}$ for $0\le t\le 1$ ....
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Let $a,b,c \le 0$. Then $\max(a,c)+\max(b,c) \le \max(a+b, c)$

I'm reading a lecture note in which the author uses Let $a,b,c \le 0$. Then $$\max(a,c)+\max(b,c) \le \max(a+b, c).$$ Below is my attempt. Is there other way to look at the problem and have a more ...
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Exercise 1, Section 32 of Munkres’ Topology

Show that a closed subspace of a normal space is normal. My attempt: Approach(1): Let $\{y\}$ be a singleton set in $Y$. Since $X$ is $T_1$, $\{y\}$ is closed in $X$. $\{y\} =Y\cap \{y\}$. By theorem ...
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Show that $SL_2(F_3)/Z(SL_2(F_3)) \cong A_4$
Show that $SL_2(F_3)/Z(SL_2(F_3)) \cong A_4$ I know that $|SL_2(F_3)/Z(SL_2(F_3))|= 12$. Then if the quotient group has a normal subgroup of order $4$ then it is isomorphic to $A_4$. Suppose that it ...