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

learn more… | top users | synonyms

50
votes
9answers
2k 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, ...
31
votes
1answer
2k 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 ...
28
votes
3answers
535 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 ...
27
votes
3answers
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 ...
19
votes
9answers
3k views

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

Maybe enough so to explain it to children.
18
votes
1answer
363 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 ...
17
votes
2answers
654 views

Prove that $x^3 + y^3 = z^3$ has no integer solutions as simply as possible

Can someone prove 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 simply as possible? I have seen some good proofs, but ...
17
votes
3answers
658 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 ...
16
votes
5answers
473 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 ...
16
votes
2answers
347 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 ...
13
votes
1answer
244 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 ...
13
votes
1answer
1k 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 ...
12
votes
3answers
283 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 ...
12
votes
1answer
419 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 ...
11
votes
4answers
193 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 ...
10
votes
4answers
269 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 ...
9
votes
7answers
869 views

If $xy$ and $x+y$ are both even integers (with $x,y$ integers), then $x$ and $y$ are both even integers

The title statement can be proven using the contrapositive, note that $x$ odd or $y$ odd means that at least one of $x\cdot y,x+y$ is odd. Is there a way to prove the statement directly? To ...
9
votes
1answer
151 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)} - ...
9
votes
3answers
352 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 ...
9
votes
2answers
690 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 ...
9
votes
1answer
604 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 ...
9
votes
4answers
244 views

How to prove $f\equiv 0$ without Weierstrass theorem?

Let $\,f:[0,1] \to \mathbb{R}$ continuous. Show that: If $$\int_0 ^1 x^k f(x)\, dx=0,$$ for all $k\in\mathbb N$, then $f\equiv 0$. I know we can use Weierstrass theorem but I'd like to ...
9
votes
3answers
127 views

Please review my question and solution. Thanks in advance.

How many values of x are there such that there exists positive integer solutions for S, such that $S=\sqrt{x(x+p)}$ where $x$ is an integer and $p$ is a prime number $>2$ This is a problem I made ...
9
votes
0answers
215 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 ...
8
votes
1answer
1k 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: ...
8
votes
1answer
861 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 ...
8
votes
2answers
202 views

What is it that makes this proof about rational rectangles work fundamentally?

I saw this problem several years ago, and I discovered a solution to it. I've since learned a somewhat more efficient solution based on the same idea. Call a rectangle in the $(x,y)$ plane ...
7
votes
11answers
804 views

limit question: $\lim\limits_{n\to \infty } \frac{n}{2^n}=0$

$$ \lim_{n\to\infty}\frac n{2^n}=0. $$ I know how to prove it by using the trick, $2^n=(1+1)^n=1+n+\frac{n(n-1)}{2}+\text{...}$ But how to prove it without using this?
7
votes
5answers
562 views

Proving $\sum_{k=1}^n k\cdot k! = (n+1)!-1$ without using mathematical Induction. [duplicate]

Possible Duplicate: Summation of a factorial This equation is given: $$ 1\cdot1! + 2\cdot2! + 3\cdot3! + \ldots + n\cdot n! = (n+1)! - 1 $$ I've solved it using mathematical induction but ...
7
votes
2answers
481 views

“Proof” that $\mathbb{R}^J$ is not normal when $J$ is uncountable

In 'Topology' by Munkres, he leaves as an exercise to prove that $\mathbb{R}^J$ is not normal under the product topology when $J$ is uncountable. The proof outlined as exercise 32.9 is the same one ...
7
votes
1answer
127 views

Direct combinatorial proof of a sum identity on formal Lagrange polynomials

Let $k$ be a field and $K=k(x_0,x_1,\ldots, x_n)[x]$. Define $$\mathcal{L}_k(x)\triangleq \prod_{\substack{j=0\\ j\ne k}}^n\frac{x-x_j}{x_k-x_j}.$$ Is there a purely combinatorial way to show ...
6
votes
11answers
776 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 beside L'Hospital's rule.
6
votes
1answer
649 views

How to deduce open mapping theorem from closed graph theorem?

These two theorems are equivalent but I can not figure out how to deduce the open mapping from the closed graph. Can anyone give a hint or some reference?
6
votes
5answers
210 views

Different ways to prove $\sqrt p$ irrational for $p$ prime.

I know this fact can be proved by contradiction(reductio ad absurdum) but please give proofs by different methods.
6
votes
1answer
232 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 ...
6
votes
1answer
373 views

Does law of large numbers converge in $L^1$?

I've seen the law of large numbers stated mainly in two (or three) forms: $S_n/n$ converges in probability (weak law) and converges almost surely (strong law). Also, there is convergence in the ...
6
votes
3answers
1k views

'Every open set in $\mathbb{R}$ is the union of disjoint open intervals.' How do you prove this without indexing intervals with $\mathbb{Q}$?

In my book's exercises section I am asked to prove that every bounded, open set in $\mathbb{R}$ is the union of disjoin open intervals. Looking around the internet I have found many strategies that ...
6
votes
1answer
3k views

Sum of irrational numbers

Well, in this question it is said that $\sqrt[100]{\sqrt3 + \sqrt2} + \sqrt[100]{\sqrt3 - \sqrt2}$, and the owner asks for "alternative proofs" which do not use rational root theorem. I wrote an ...
6
votes
1answer
268 views

Proof that a certain entire function is a polynomial

Let $n\in\mathbf{N}$ be fixed, and $f$ entire and $|f^{-1}(\left\lbrace w\right\rbrace)|\leq n$ for every $w\in\mathbf{C}$. Then $f$ is a polynomial of degree at most $n$. I try to prove this ...
6
votes
1answer
148 views

Combinatorial proof of the fact $p$ doesn't divide $ n \choose p^k$

Let $p^k | n$ and $p^{k+1} \nmid n$. Is there any combinatorial proof of the fact that $p \nmid {n \choose p^{k}} $ ?
6
votes
1answer
115 views

Uniform Convergence verification for Sequence of functions - NBHM

Following is a list of problems from an exam for admission into Ph.D program. I have just compiled all previous questions on uniform convergence of sequence of functions and i tried to work out . I ...
6
votes
0answers
201 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 know that Choice principles have some connection to the separation axioms (in ZF, at ...
5
votes
4answers
201 views

Prove the following identity

I am having some trouble proving following identity without use of induction, with which it is trivial. $$\sum_{n=1}^{m}\frac{1}{n(n+1)(n+2)}=\frac{1}{4}-\frac{1}{2(m+1)(m+2)}$$ I did expand the ...
5
votes
2answers
83 views

Cool property of the number $24$

Recently I've had my 24th birthday, and a friend commented that it was a very boring number, going from 23 which is prime, 25 which is the first number that can be written as the sum of 2 different ...
5
votes
2answers
215 views

Is there a proof of the irrationality of $\sqrt{2}$ that involves modular arithmetic?

I was reading Ian Stewart's Concepts of Modern Mathematics. Using congruences, It's possible to explain why all perfect squares end in $0,1,4,5,6,9$ but not in $2,3,7,8$. With this I had the ...
5
votes
5answers
538 views

Proving $\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$ without induction [duplicate]

I was looking at: $$\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$$ It's pretty easy proving the above using induction, but I was wondering what is the actual way of getting this equation?
5
votes
5answers
163 views

interesting Integral , alternative solution.

Show the following relation: $$\int_{0}^{\infty} \frac{x^{29}}{(5x^2+49)^{17}} \,\mathrm dx = \frac{14!}{2\cdot 49^2 \cdot 5^{15 }\cdot 16!}.$$ I came across this intgeral on a physics forum and ...
5
votes
3answers
284 views

On Ceva's Theorem?

The famous Ceva's Theorem on a triangle $\Delta \text{ABC}$ $$\frac{AJ}{JB} \cdot \frac{BI}{IC} \cdot \frac{CK}{EK} = 1$$ is usually proven using the property that the area of a triangle of ...
5
votes
1answer
313 views

Excessive use of the Yoneda lemma

In a MathOverflow thread on "nuking mosquitos", Andrej Bauer offered the following proof: If two elements in a poset have the same lower bounds then they are equal by Yoneda lemma. I understand ...
5
votes
1answer
214 views

Using Nakayama's Lemma to prove isomorphism theorem for finitely generated free modules

Suppose $R \neq 0$ is a commutative ring with $1$. The following is well known: (Isomorphism Theorem for Finitely Generated Free Modules) [FGFM] $R^{n}\cong R^{m}$ as $R$-modules if and only if ...