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

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26
votes
9answers
995 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 ...
0
votes
1answer
136 views

On a certain basis of an order of a quadratic number field

Let $K$ be an algebraic number field of degree $n$. An order of $K$ is a subring $R$ of $K$ such that $R$ is a free $\mathbb{Z}$-module of rank $n$. Let $\alpha_1, \cdots, \alpha_n$ be a basis of $R$ ...
17
votes
5answers
629 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 ...
4
votes
3answers
923 views

$p=4n+3$ never has a Decomposition into $2$ Squares, right?

Primes of the form $p=4k+1\;$ have a unique decomposition as sum of squares $p=a^2+b^2$ with $0<a<b\;$, due to Thue's Lemma. Is it correct to say that, primes of the form $p=4n+3$, never have ...
7
votes
1answer
392 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 ...
2
votes
1answer
289 views

Another point of view that $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ is cyclic.

I was thinking that the product of groups $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/2\mathbb{Z}$ is not cyclic, but $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/p\mathbb{Z}$ is cyclic if p is an odd prime. ...
3
votes
1answer
135 views

How to show that $\lfloor n/1\rfloor+\lfloor n/2 \rfloor+…+\lfloor n/n\rfloor+\lfloor{\sqrt{n}}\rfloor$ is even?

Let $n$ is a natural number. Prove that $$\left\lfloor\frac{n}{1}\right\rfloor+\left\lfloor\frac{n}{2}\right\rfloor+....+\left\lfloor\frac{n}{n}\right\rfloor+\left\lfloor{\sqrt{n}}\right\rfloor$$ ...
7
votes
5answers
611 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 ...
-4
votes
1answer
155 views

Canonical basis of an ideal of a quadratic order

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, $D$ its discriminant. I am interested in the ideal theory on $R$ because it is closely related to the theory of binary quadratic forms ...
12
votes
3answers
485 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 ...
-1
votes
1answer
83 views

Condition for $a, b + \omega$ to be the canonical basis of an ideal of a quadratic order

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, $D$ its discriminant. By this question, $1, \omega = \frac{(D + \sqrt D)}{2}$ is a basis of $R$ as a $\mathbb{Z}$-module. Let $I$ be a ...
13
votes
1answer
1k 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 ...
10
votes
4answers
385 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 that it can be proved using Weierstrass Theorem, ...
6
votes
3answers
648 views

Is this proof that $\sqrt 2$ is irrational correct?

Suppose $\sqrt 2$ were rational. Then we would have integers $a$ and $b$ with $\sqrt 2 = \frac ab$ and $a$ and $b$ relatively prime. Since $\gcd(a,b)=1$, we have $\gcd(a^2, b^2)=1$, and the fraction ...
-2
votes
1answer
237 views

Criterion on whether a given ideal of a quadratic order is regular or not

Let $K$ be an algebraic number field of degree $n$. Let $\mathcal{O}_K$ be the ring of algebraic integers. Let $R$ be an order of $K$, i.e. a subring of $K$ which is a free $\mathbb{Z}$-module of rank ...
35
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 ...
19
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 ...
30
votes
3answers
787 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 ...
0
votes
3answers
1k views

for $1<p<2$, prove the p-series is convergent without concerned with integral and differential knowledge and geometry series

for $1<p<2$, prove the p-series: $\sum_{n=1}^{\infty}n^{-p}$ is convergent. please use Cauchy Rule (edit: that is, by showing directly that the sequence of partial sums is a Cauchy sequence) ...
14
votes
6answers
580 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, ...
8
votes
5answers
1k 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? ...
5
votes
1answer
404 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 ...
1
vote
2answers
783 views

Using inequalities and limits

Is it possible to say: $$ If \ f(x) \ and \ g(x) \ both \ have \ limits \ as \ x\to p\ and \ f(x) \le g(x), \ then \lim_{x \to p} f(x)\le \lim_{x \to p} g(x). $$ My proof(Edit: Proof is wrong due to ...
11
votes
0answers
271 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 ...
2
votes
8answers
436 views

Why Not Define $0/0$ To Be $0$?

For every number $x$, $x\times 0=0$, hence $\dfrac{0}{0}$ can be any number! So $\dfrac{0}{0}$ "is knows as indeterminate" [1]. But what if we define it to be $0$? I already have an answer, but ...
0
votes
4answers
3k views

Prove that a continuous function on a closed interval attains a maximum

As the title indicates, I'd like to prove the following: If $f:\mathbb R\to\mathbb R$ is a continuous function on $[a,b]$, then $f$ attains its maximum. Now, I do have a working proof: $[a,b]$ ...
18
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 ...
16
votes
10answers
3k 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 ...
4
votes
4answers
2k views

Insightful proofs for Sherman-Morrison Formula and Matrix Determinant Lemma

There are two statement about a matrix under rank-one updates that I would be grateful if you give me some insightful proofs. Suppose $A$ be a nonsingular $n \times n$ matrix and ...
6
votes
3answers
94 views

Prove $(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}$.

I can show that for $x > 0$ and $r_{i} > 0$ we have $$ \left(\, x + r_{1}\,\right)\ldots\left(\, x + r_{n}\,\right)\ \geq\ \left[\, x + \left(\, r_{1}\ldots r_{n}\,\right)^{1/n}\,\right]^{n}.$$ ...
4
votes
1answer
188 views

Identity concerning complete elliptic integrals

It can be easily checked that both the complete elliptic integrals $K(k), K'(k)$ satisfy the same second order differential equation $$kk'^{2}\frac{d^{2}y}{dk^{2}} + (1 - 3k^{2})\frac{dy}{dk} - ky = ...
5
votes
1answer
80 views

All $k$-regular subgraphs of $K_{n,n}$ have a perfect matching: a proof without Hall's Marriage Theorem?

There are several ways of describing this result: Theorem: For $k \in \{1,2,\ldots,n\}$, any $k$-regular subgraph of $K_{n,n}$ has a perfect matching (also known as a $1$-factor). I tend to ...
5
votes
2answers
240 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 ...
4
votes
6answers
166 views

Prove that $x-1$ is a factor of $x^n-1$

Prove that $x-1$ is a factor of $x^n-1$. My problem: I already proved it by factor theorem† and by simply dividing them. I need another approach to prove it. Is there any other third ...
4
votes
2answers
177 views

If $n$ is squarefree, $k\ge 2$, then $\exists f\in\Bbb Z[x_1,\dots,x_k] : f(\overline x)\equiv 0\pmod n\iff \overline x\equiv \overline 0\pmod n$

Starting from this question, we set $n=k=2$ and use the function $f\in\Bbb Z[x,y]$ where $f(x,y)=x\cdot y+x+y$, then the proofs applied to that question satisfy this case. Note that for $k=1$ the ...
3
votes
2answers
105 views

Characterize the natural numbers $n$ such that there is a surjective group homomorphism from $S_n$ to $S_{n-1}$.

This has always been one of my most favorite exercises from Group Theory, and I was surprised to see that this hasn't been asked before. To repeat: Characterize the natural numbers $n$ such that ...
3
votes
5answers
208 views

$\sum_{k=1}^nH_k = (n+1)H_n-n$. Why?

This is motivated by my answer to this question. The Wikipedia entry on harmonic numbers gives the following identity: $$ \sum_{k=1}^nH_k=(n+1)H_n-n $$ Why is this? Note that I don't just ...
3
votes
2answers
117 views

Prove: if $f(x) =x\sin (\pi x)$ then $f'(x)$ vanishes at a unique point in $ ( n + 1/2, n) $

Let $ f(x) = x\sin (\pi x), x > 0 $. Then prove that for all natural numbers n, $f'(x)$ vanishes at a unique point in $ ( n + 1/2, n) $ The given solution shows a graph, but is there any algebraic ...
2
votes
1answer
174 views

Equivalences to “D-finite = finite”

By a D-finite set, we mean a set admitting no injection from the natural numbers (or equivalently, a set not in bijection with any proper subset). I have encountered a proof that the following are ...
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) ...
4
votes
0answers
358 views

Equivalence of Brouwers fixed point theorem and Sperner's lemma

I'm looking for a proof that Brouwer's fixed point theorem implies Sperner's lemma. All proof I've found just prove that there must be at least one completely colored n-simplex, not that there must be ...
3
votes
4answers
824 views

Axiom of Choice and Right Inverse

I read an Theorem that states: Let $A$ and $B$ be non-empty sets, and let $f:A \to B$ be a function, then the function $f$ has a right inverse if and only if $f$ is surjective. The Theorem ...
2
votes
2answers
36 views

Next step to take to reach the contradiction?

This problem is from Discrete Math and its Applications I am trying to use proof by contradiction to do this problem, proof by contradiction as described by the book Here is my work so far for ...
2
votes
4answers
408 views

Prove $1 + \tan^2\theta = \sec^2\theta$

Prove the following trigonometric identity: $$1 + \tan^2\theta = \sec^2\theta$$ I'm curious to know of the different ways of proving this depending on different characterizations of tangent and ...
2
votes
1answer
115 views

Factor ring by a regular ideal of a one-dimensional Noetherian domain

Let $A$ be a one-dimensional Noetherian domain. Let $K$ be its field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is a finitely generated $A$-module. It is well-known that ...
1
vote
3answers
455 views

Proof of the duality of the dominance order on partitions

Could anyone provide me with a nice proof that the dominance order $\leq$ on partitions of an integer $n$ satisfies the following: if $\lambda, \tau$ are 2 partitions of $n$, then $\lambda \leq \tau ...
0
votes
1answer
44 views

Did I do this big-Omega proof correctly?

Prove or disprove: 6n^3 – 4n^2 + 3n +2 is in Ω (5n^3 – n^2 + n +1). So I'm not sure if I did this right or not, any pointers or the correct steps would be helpful Ǝc ∈ ℝ+, ƎB ∈ ℕ, ∀n ∈ ℕ, n ≥ B ⇒ ...
0
votes
1answer
178 views

Proving isomorphism between between a subspace and a quotient space

I've been thinking about this for a day or two now, and I think I've found a way to prove this, but am very unsure about how watertight this is: To be proven: Let $V$ be a vector space. If $V = U ...
0
votes
1answer
116 views

Decomposition of a primitive ideal of a quadratic order

Let $K$ be a quadratic number field. Let $\mathcal{O}_K$ be the ring of algebraic integers in $K$. Let $R$ be an order of $K$, $D$ its discriminant. I am interested in the ideal theory on $R$ because ...
0
votes
1answer
175 views

On the norm formula $N(IJ) = N(I)N(J)$ in an order of an algebraic number field

Let $K$ be an algebraic number field of degree $n$. Let $\mathcal{O}_K$ be the ring of algebraic integers of $K$. Let $R$ be an order of $K$, i.e. a subring of $K$ which is a free $\mathbb{Z}$-module ...