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

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88
votes
1answer
4k views

$n!$ 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 $...
59
votes
15answers
3k views

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 ...
59
votes
9answers
3k views

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, ...
58
votes
2answers
1k views

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 ...
44
votes
13answers
4k views

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 ...
38
votes
3answers
1k views

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....
33
votes
12answers
889 views

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 ...
30
votes
3answers
5k 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 ...
30
votes
3answers
1k views

**Ended Competition:** What is the shortest proof of $\exists x \forall y (D(x) \to D(y)) $?

The competition has ended 6 june 2014 22:00 GMT The winner is Bryan Well done ! When I was rereading the proof of the drinkers paradox (see Proof of Drinker paradox I realised that $\exists x \...
25
votes
1answer
291 views

Is there an easy way to see that this simple recurrence is 9-periodic?

In a colloquium talk yesterday, Robert Bryant pointed out that for all initial values $a_0, a_1 \in \mathbb{R}$, the sequence generated by the recurrence relation $$ a_{n+1} = |a_n| - a_{n-1} $$ turns ...
25
votes
1answer
788 views

Does Nakayama Lemma imply Cayley-Hamilton Theorem?

Consider the Cayley-Hamilton Theorem in the following form: CH: Let $A$ be a commutative ring, $\mathfrak{a}$ an ideal of $A$, $M$ a finitely generated $A$-module, $\phi$ an $A$-module endomorphism ...
24
votes
7answers
1k views

A limit problem $\lim\limits_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$

This is a problem from "A Course of Pure Mathematics" by G H Hardy. Find the limit $$\lim_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$$ I had solved it long back (solution presented in my blog ...
22
votes
1answer
266 views

$\log_2 13$ is irrational

Is it true that $\log_2 13$ is irrational? Let $x=\log_2 13\implies 2^x=13$. So, it will be an irrational number, if not,$$x=\frac p q$$ and $$2^{\frac p q}=13$$ $$\implies 2^p=13^{q}$$ Since, $...
21
votes
4answers
2k views

Proofs that every natural number is a sum of four squares.

I am planning to write a little note detailing several proofs of Lagrange's theorem that every natural number can be written as the sum of four perfect squares. I know of three different proofs so far:...
21
votes
4answers
4k views

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 ...
21
votes
1answer
511 views

Verify matrix identity $A^tD-C^tB=I$ on certain hypotheses

Given $n\times n$ real matrices $A,B,C,D$ such that: $AB^T$ and $CD^T$ are symmetric $AD^T-BC^T=I$ Prove that $A^TD-C^TB=I$ The solution I have come up with after a very long time is to consider: ...
20
votes
9answers
14k views

What is the simplest proof of the pythagorean theorem you know? [duplicate]

Maybe enough so to explain it to children.
19
votes
3answers
891 views

A Geometric Proof of $\zeta(2)=\frac{\pi^2}6$? (and other integer inputs for the Zeta)

Is there a known geometric proof for this famous problem? $$\zeta(2)=\sum_{n=1}^\infty n^{-2}=\frac16\pi^2$$ Moreover we can consider possibilities of geometric proofs of the following identity for ...
19
votes
1answer
2k views

A proof of Wolstenholme's theorem

This was inspired by this question. I tried to use the identity $${2n \choose n}=\sum_{k=0}^n {n \choose k}^2$$ (see this question) to prove that $$\binom{2p}p\equiv2\pmod{p^3}$$ if $p\gt3$ is prime....
19
votes
1answer
429 views

Easy proof, that $\rm e\notin \mathbb Q$

$\def\e{{\rm e}}$ I recently had the task to explain the proof that $\e$ is irrational as a presentation to my classmates. To prepare this presentation, the teacher gave me a script with a proof that ...
18
votes
1answer
487 views

“Novel” proofs of “old” calculus theorems

Every once in a while some mathematicians publish (mostly on the American Mathematical Monthly) a new proof of an old (nowadays considered "basic") result in analysis (calculus). This article is an ...
17
votes
1answer
137 views

If the Greeks had been four dimensional, would they have been able to derive the pi squared coefficient for the hypersphere volume without calculus?

I was reading about Archimedes' pre-calculus proof of the volume of the sphere and I realized that the trick he uses (volume of hemisphere + volume of cone = volume of cylinder) doesn't generalize to ...
16
votes
10answers
5k views

Polygons with equal area and perimeter but different number of sides?

Let's say we have two polygons with different numbers of sides. They can be any sort of shape, but they have to have the same area, and perimeter. There could be such possibilities, but can someone ...
16
votes
9answers
425 views

Lesser known derivations of well-known formulas and theorems

What are some lesser known derivations of well-known formulas and theorems? I ask because I recently found a new way to derive the quadratic formula which didn't involve completing the square as is ...
16
votes
5answers
488 views

Visualization of surface area of a sphere

I help mentor some really young, bright kids in mathematics. We were looking at geometric properties of various shapes, and one of the kids noted that the surface area of a sphere $S = 4\pi r^2$ ...
16
votes
2answers
2k views

A snappy proof of Fatou's lemma

I'm studying basic real analysis and stumbled across three big results (Fatou's lemma, Lebesgue's Dominated Convergence theorem, and the Monotone Convergence theorem) in the theory of Lebesgue ...
16
votes
2answers
489 views

Writing a group element as $ghg^{-1} h^{-1}$ and as $g^2 h^2$

I recently read the elegant paper Generalized Frobenius Schur Numbers, by Bump and Ginzburg, which I learned about here. The results in this paper imply the following: Let $G$ be a finite group ...
16
votes
2answers
904 views

A simply-connected closed surface is a sphere

From the Classification Theorem for closed (i.e. compact and boundaryless) surfaces, it follows that $S^2$ is the only closed surface with trivial $\pi _1$. That's easy because the fundamental group ...
15
votes
5answers
2k views

Is there an idempotent element in a finite semigroup?

Let $(G,\cdot)$ be a finite semigroup. Is there any $a\in G$ such that: $$a^2=a$$ It seems to be true in view of theorem 2.2.1 page 97 of this book (I'm not sure). But is there an elementary proof? ...
15
votes
1answer
2k views

Direct aproach to the Closed Graph Theorem

In the context of Banach spaces, the Closed Graph Theorem and the Open Mapping Theorem are equivalent. It seems that usually one proves the Open Mapping Theorem using the Baire Category Theorem, and ...
15
votes
6answers
794 views

Fascinating induction problem with numerous interpretations

Problem: Suppose you begin with a pile of $n$ stones and split this pile into $n$ piles of one stone each by successively splitting a pile of stones into two smaller piles. Each time you split a pile, ...
15
votes
4answers
679 views

Easy proof for sum of squares $\approx n^3/3$

I'd like to prove to my (undergraduate, not math-major) students that $$ \lim_{n\to\infty} \frac{1}{n^3}\sum_{k=1}^n k^2 =\frac{1}{3}, $$ to later show them that this can be interpreted as taking ...
14
votes
0answers
314 views

Pólya and Szegő, Part I, Ch. 4, 174.

The following is a problem proposed in Pólya and Szegő's book "Problems and Theorems in Analysis" Assume that $0<f(x)<x$ and $$f(x)=x-ax^k+bx^\ell+x^\ell \varepsilon(x),\,\;\;\;\lim_{x\to 0}\...
13
votes
1answer
368 views

A beautiful inequality for convex functions

Let $f\in \mathcal{C}([0,1],\mathbb R_+)$ increasing. Prove that there exist $g,h\in \mathcal{C}([0,1],\mathbb R)$, convexs, such that $g\leqslant f \leqslant h$ and : $$\displaystyle \frac{1}...
12
votes
3answers
2k views

Show that the product of two consecutive natural numbers is never a square.

I'd like to have my proof verified and if possible, to see other solutions that are interesting. Proof: Suppose $n(n+1)$ is a square. Then we write $$n(n+1) = \prod_{p} p^{c(p)}$$ where $c(p) = a(p) +...
12
votes
3answers
643 views

Can $\displaystyle\lim_{h\to 0}\frac{b^h - 1}{h}$ be solved without circular reasoning?

In many places I have read that $$\lim_{h\to 0}\frac{b^h - 1}{h}$$ is by definition $\ln(b)$. Does that mean that this is unsolvable without using that fact or a related/derived one? I can of course ...
12
votes
1answer
327 views

Alternative proof of simple integral inequality

Problem. Let $f\in C^1(\mathbb R)$ such that $f(0) = 0$ and $0 < f'(x) \le 1$. Prove that for all $x\ge 0$ $$ \int_0^x f^3(t)\,dt \le \left(\int_0^x f(t)\,dt\right)^{\!\!2}. $$ Below is my ...
12
votes
3answers
305 views

Simplified form for $\frac{\operatorname d^n}{\operatorname dx^n}\left(\frac{x}{e^x-1}\right)$?

I have found the following formula: $$\frac{\operatorname d^n}{\operatorname dx^n}\left(\frac{x}{e^x-1}\right)=(-1)^n\,\frac{n\sum\limits_{k=0}^{n}e^{kx}\sum\limits_{i=0}^{k}(-1)^i\binom{n+1}{i}(k-i)^...
12
votes
1answer
961 views

Hahn-Banach theorem: 2 versions

I have a question regarding the Hahn-Banach Theorem. Let the analytical version be defined as: Let $E$ be a vector space, $p: E \rightarrow \mathbb{R}$ be a sublinear function and $F$ be a subspace of ...
12
votes
5answers
327 views

Checking that a $3$-D diagram is commutative

When proving certain results I need to use commutative diagrams, some of which quite complicated. My question is: Do we need to check every small square all the time to make sure that they are all ...
12
votes
1answer
320 views

Seeking a more direct proof for: $m+n\mid f(m)+f(n)\implies m-n\mid f(m)-f(n)$

If $f:\mathbb N\to\mathbb Z$ satisfies: $$\forall n,m\in\mathbb N\,, n+m\mid f(n)+f(m)$$ How to show that this implies: $$\forall n,m\in\mathbb N,\,n-m\mid f(n)-f(m)?$$ I was almost incidentally ...
11
votes
1answer
3k views

Solving SAT by converting to disjunctive normal form

The first well-known $NPC$ problem is the Boolean Satisfiability Problem, which has a proof of being $NPC$ done by Cook (Cook-Levin Theorem). The problem can easily be described the following way: ...
11
votes
2answers
1k views

Algebraic proof of a trig matrix identity?

I'll put the question first, and then the background, because I'm not sure that the background is necessary to answer the question: I have a geometric proof, but is there an elegant algebraic proof ...
11
votes
4answers
2k views

How can I complete this proof by contradiction?

This problem is from Discrete Mathematics and its Applications: Prove that there are no solutions in integers $x$ and $y$ to the equation $2x^2 + 5y^2 = 14$. I am trying to use proof by ...
11
votes
1answer
584 views

The ring of integers of a number field is finitely generated.

For a number field $K$, we define the ring of integers of $K$ to be $$\mathcal{O}_K:=\{x\in K\big|\ (\exists f\in\mathbb{Z}[X])(f\ \text{ is monic and } f(x)=0)\}.$$ Is there any easy way to see from ...
11
votes
1answer
217 views

Does the functional equation $p(x^2)=p(x)p(x+1)$ have a combinatorial interpretation?

A recent question asked about polynomial solutions to the functional equation $p(x^2)=p(x)p(x+1)$. Subsequently, Robert Israel posted an answer showing that solutions are necessarily of the form $p(x)=...
10
votes
12answers
2k views

How to prove that $\lim\limits_{x\to0}\frac{\tan x}x=1$?

How to prove that $$\lim\limits_{x\to0}\frac{\tan x}x=1?$$ I'm looking for a method besides L'Hospital's rule.
10
votes
4answers
234 views

Inequality $\binom{2n}{n}\leq 4^n$

I would like to prove the following inequality, for $n=0,1,2,...$, $$ \binom{2n}{n}\leq 4^n.$$ I already proved it by induction, and I'm looking for another proof.
10
votes
1answer
215 views

Is my proof correct for: $\sqrt[7]{7!} < \sqrt[8]{8!}$

I have to show that $$\sqrt[7]{7!} < \sqrt[8]{8!}$$ and I did the following steps \begin{align} \sqrt[7]{7!} &< \sqrt[8]{8!} \\ (7!)^{(1/7)} &< (8!)^{(1/8)} \\ (7!)^{(1/7)} - (...
10
votes
4answers
493 views

Construct quadrangle with given angles and perpendicular diagonals

The following came up when I worked on the answer for a different question (though it was ultimately not used in this form): Proposition. Given positive angles $\alpha,\beta,\gamma,\delta$ with $\...