If you already have a proof for some result but want to ask for a different proof (using different methods).
Questions tagged [alternative-proof]
3799 questions
138
votes
30 answers
Proofs of AM-GM inequality
The arithmetic - geometric mean inequality states that
$$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$
I'm looking for some original proofs of this inequality. I can find the usual proofs on the internet but I was wondering if someone…
Michiel Van Couwenberghe
- 2,847
124
votes
20 answers
What is the most unusual proof you know that $\sqrt{2}$ is irrational?
What is the most unusual proof you know that $\sqrt{2}$ is irrational?
Here is my favorite:
Theorem: $\sqrt{2}$ is irrational.
Proof:
$3^2-2\cdot 2^2 = 1$.
(That's it)
That is a corollary of
this result:
Theorem:
If $n$ is a positive…
marty cohen
- 110,450
119
votes
1 answer
$n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem?
If $n\geq2$, then $n!$ is not a perfect square. The proof of this follows easily from Chebyshev's theorem, which states that for any positive integer $n$ there exists a prime strictly between $n$ and $2n-2$. A proof can be found here.
Two weeks and…
Lincoln Blackham
- 1,191
82
votes
2 answers
Visual proof of $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$?
In his gorgeous paper "How to compute $\sum \frac{1}{n^2}$ by solving triangles", Mikael Passare offers this idea for proving $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$:
Proof of equality of square and curved areas is based on another…
VividD
- 16,196
71
votes
11 answers
Is it technically incorrect to write proofs forward?
A question on an assignment was similar to prove:
$$2a^2-7ab+2b^2 \geq -3ab.$$
and my proof was:
$$2a^2-4ab+2b^2\geq0$$
$$a^2-2ab+b^2\geq0$$
$$(a-b)^2\geq0$$
which is true.
However, my professor marked this as incorrect and the "correct" way to do…
mtheorylord
- 4,340
67
votes
1 answer
Proving that $\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$
Prove without evaluating the integrals that:$$2\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx=\int_\frac{\pi}{2}^\pi\frac{x\ln(1-\sin x)}{\sin x}dx\label{*}\tag{*}$$
Or equivalently:
$$\boxed{\int_0^\pi\frac{x\ln(1-\sin x)}{\sin…
Zacky
- 30,116
60
votes
9 answers
Surprisingly elementary and direct proofs
What are some examples of theorems, whose first proof was quite hard and sophisticated, perhaps using some other deep theorems of some theory, before years later surprisingly a quite elementary, direct, perhaps even short proof has been found?
A…
Martin Brandenburg
- 181,922
51
votes
4 answers
Three semicircles geometry problem from TikTok
Here's a fun geometry problem I stumbled on TikTok. We are given the diameters of the red and yellow circles (which are tangent) and supposed to determine the diameter of the big circle.
It is easy to trace some triangles, apply Pythagoras and find…
Alma Arjuna
- 6,521
51
votes
5 answers
Alternative proofs that $A_5$ is simple
What different ways are there to prove that the group $A_5$ is simple?
I've collected these so far:
By directly working with the cycles: page 483 of http://www.math.uiowa.edu/~goodman/algebrabook.dir/algebrabook.html
Because it has order 60 and two…
user58512
50
votes
3 answers
A $\{0,1\}$-matrix with positive spectrum must have all eigenvalues equal to $1$
Here's a cute problem that was frequently given by the late Herbert Wilf during his talks.
Problem: Let $A$ be an $n \times n$ matrix with entries from $\{0,1\}$ having all positive eigenvalues. Prove that all of the eigenvalues of $A$ are…
Potato
- 41,411
42
votes
1 answer
Can you make the triviality of $\langle a,b,c \mid aba^{-1} = b^2, bcb^{-1} = c^2, cac^{-1} = a^2 \rangle$ more trivial?
I recently learned the following pleasant fact. (It was in the proof of Proposition 3.1 of this paper - but don't worry, there's no model theory in this question.)
Let $G$ be a group, and let $a,b,c\in G$. If $aba^{-1} = b^2$, $bcb^{-1} = c^2$, and…
Alex Kruckman
- 86,811
42
votes
13 answers
What are the theorems in mathematics which can be proved using completely different ideas?
I would like to know about theorems which can give different proofs using completely different techniques.
Motivation:
When I read from the book Proof from the Book, I saw there were many many proof for the same theorem using completely…
Bumblebee
- 18,974
41
votes
5 answers
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 divisor of non-adjacent vertices is constant.
Below is…
milcak
- 4,199
40
votes
4 answers
The equation $x^3 + y^3 = z^3$ has no integer solutions - A short proof
Can someone provide the proof of the special case of Fermat's Last Theorem for $n=3$, i.e., that
$$
x^3 + y^3 = z^3,
$$
has no positive integer solutions, as briefly as possible?
I have seen some good proofs, but they are quite long (longer than…
qwr
- 11,362
40
votes
14 answers
Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula
As far as every question I've seen concerning "what is $\sum_{k=1}^nk^p$" is always answered with "Faulhaber's formula" and that is just about the only answer. In an attempt to make more interesting answers, I ask that this question concern the…
Simply Beautiful Art
- 76,603