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

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1answer
37 views

Proving that $z^4-6z^2+4z-3 = y^2$ has only one integer solution

I'm trying to prove the following result. Conjecture. If $z$ is an integer, and $z^4-6z^2+4z-3$ is a square, then $z=3$. A quick check modulo $9$ shows that $z=9w+3$ for some integer $w$. So for ...
2
votes
2answers
82 views

square matrix $\mathbf{A}$ with $\mathbf{A}^\intercal = -\mathbf{A}$, proof $\mathbf{A}$ is not invertible.

I tried proving that given a square $\mathbf{A}$ over $\mathbb{R}$ so that $\mathbf{A}^\intercal = -\mathbf{A}$, $\mathbf{A}$ is not invertible. I know that because the matrix is real and its ...
1
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0answers
8 views

How to prove mathematically these two different definitions of background-position property as equivalent?

I've been reading about how percentage values work for background-position position property. The official definition is that ...
5
votes
0answers
95 views

Looking for the most elementary proof that $48X^4+12X^2+1=Y^2$ has no non-trivial integer solution.

As relayed in this question of mine (which is more general in scope), I believe I have found a relatively easy, and completely elementary, way to show that the equation $$48X^4 + 12X^2+1 = Y^2$$ has ...
11
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1answer
212 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 ...
2
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0answers
38 views

Proving (without using complex numbers) that a real polynomial has a quadratic factor

The Fundamental Theorem of Algebra tells us that any polynomial with real coefficients can be written as a product of linear factors over $\mathbb{C}$. If we don't want to use $\mathbb{C}$, the best ...
0
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1answer
303 views

Is there a more rigorous way to show these two sums are exactly equal?

I would like to have a more rigorous proof of the hypothesis: The Crandall eta derivative series is equal to the following more elementary one. The following two series give the same sum. $$-\sum ...
1
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0answers
43 views

Linear Algebra Friedberg Th1.9

I have been trying to prove Th1.9 rigorously since Friedberg didn't do so. Here is my attempt at a rigorous proof. Th 1.9 Restated: If a vector space V, S has n elements and span(S)=V then some ...
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3answers
45 views

Elegant proof that maximum of sums is, at most, sum of maximums

I'm looking for an elegant way to show that, among non-negative numbers, $$ \max \{a_1 + b_1, \dots, a_n + b_n\} \leq \max \{a_1, \dots, a_n\} + \max \{b_1, \dots, b_n\} $$ I can show that $\max ...
2
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0answers
36 views

Is there a statement inside of mathematics that is proven only with mathematical induction and with none other method?

Well, the point is that although the method of mathematical induction can be useful and is useful for proving certain statements, I somehow always like things to be proved in some other way than with ...
1
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1answer
42 views

Solving $x^e =c$ in $\mathbb{F}_{p}$

Find all solutions to the equation $x^3=7$ in $\mathbb{F}_{13},\mathbb{F}_{19}$ and $\mathbb{F}_{35}$. In An Introduction to Mathematical Cryptography (Hoffstein et al), we have that proposition ...
1
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1answer
97 views

How many elementary ways are there to prove that $\displaystyle\left. \frac d {ds}\,\frac 1 {\zeta(s)} \right|_{s=1} = 1 $?

In a comment under this answer, a user boldly asserts that there is ONLY ONE way to prove that $$ \left. \frac d {ds}\,\frac 1 {\zeta(s)} \right|_{s=1} = 1 $$ where $\zeta$ is Riemann's zeta function. ...
0
votes
1answer
60 views

Trace of matrix is sum of eigenvalues (positive semi-definite case)

Let $A \in \mathbb{R}^{n \times n}$. It is well-known that $\text{tr}(A)$ is equal to the sum of the eigenvalues of $A$. Let us know restrict $A$ to being positive semi-definite. Obviously, it is ...
2
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4answers
54 views

Proof verification : every algebraic set is the union of finitely many irreducible algebraic subsets

I have found various proofs of the result but I have come up with something very different and I wonder whether it is a valid argument: Let $W$ be an algebraic set. Let $I=\mathcal{I}(W)$. We have ...
1
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1answer
15 views

Proof improvement for $(a^2+b^2)(c^2+d^2)(e^2+f^2)(g^2+h^2) = A^2 + B^2$ given $(a+ib)(c+id)(e+if)(g+ih) = A + iB$

If $(a+ib)(c+id)(e+if)(g+ih) = A + iB$, prove that $(a^2+b^2)(c^2+d^2)(e^2+f^2)(g^2+h^2) = A^2 + B^2$ My approach is pretty straightforward: $$(a+ib)(c+id)(e+if)(g+ih)$$ ...
0
votes
1answer
36 views

If $A$ Is an Upper Triangle Matrix, the Adjoint Is Also Upper Triangular

I already proved it, but it was really laborious. I am wondering if any one has a shorter proof? Write $A = [a_{ik}]$ and let $\overline{A}_{rs} = [c_{ik}]$ denote the minor with row $r$ and column ...
4
votes
3answers
357 views

Finding the shortest distance between two Parabolas

Recently, a problem asked me to find the minimum distance between the parabolas $y=x^2$ and $y=-x^2-16x-65$. I proceeded with the problem as thus. Let $P(a,a^2), Q(b, -b^2-16b-65), a-b=x$. ...
1
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0answers
51 views

Quick way to classify groups of small order

This is a past Qualifying exam on Algerbra : I'm curious if there is a quick way to solve Prob 1. To me, directly classifying groups takes too much time and I think I could not handle this in time ...
1
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1answer
40 views

Prove $ f(c)\int_{a}^{b}g(x)dx=\int_{a}^{b}g(x)f(x)dx$

Assume that $f:[a,b]\rightarrow\mathbb{R}$ is continuous on $[a,b]$ and $g:[a,b]\rightarrow\mathbb{R}$ is integrable and $g(x)\geq0$ for all $x\in[a,b]$. Then there exists a $c\in(a,b)$ such that ...
0
votes
1answer
23 views

Alternative methods to solve DLP for $GL_{3}(\mathbb{F}_2)$

Is there (or rather what is) a more elegant/efficient way to solve the DLP for $g^x=h$ in $GL_3(\mathbb{F}_2)$ where $$g=\begin{pmatrix}0 &1 & 1 \\ 1 &1 &1 \\ 1&0&1 ...
1
vote
1answer
26 views

Extension of co-coercivity in strongly convex functions

I am studying strongly convex functions and they mention if $f(x)$ is strongly convex with Lipschitz gradients $L$, which means $\parallel \nabla f(y) - \nabla f(x)\parallel \leq L\parallel x - y ...
0
votes
1answer
36 views

Function is differentiable in all the points of its domain

I need to proof that this function is differentiable in all the points of its domain. I know that this is true if the function is a function $\in C^k$ and a function is $C^k$ if is composition of ...
4
votes
1answer
239 views

Proof that $\frac{2}{3} < \log(2) < \frac{7}{10}$

Positive integrals $$\int_{0}^{1}\frac{2x(1-x)^2}{1+x^2}dx=\pi-3$$ and $$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi $$ (http://math.stackexchange.com/a/1618454/134791) prove that ...
1
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2answers
31 views

Help Proving the Average is greater than B^(1/n)

Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Define the following two numbers: $A = (a_1 + a_2 + \cdots + a_n) /n $ (The average of the numbers) $B = (a_1 + a_2 + \cdots + ...
2
votes
2answers
56 views

Solving the recurrence $A_n = \sum_{k=1}^{n} 2^{k+1} A_{n-k}$

Let me ask a very simple question: Let $(A_n)$ be a sequence of integers defined by $A_0 = 1$ and $$\forall n \geq 1 : A_n = \sum_{k=1}^{n} 2^{k+1} \cdot A_{n-k}.$$ There is an explicit formula for ...
0
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2answers
20 views

Decreasing sequence and prove by contradiction

I have "solved" the following question using prove by contradiction. But it seems a bit off to me: Let {$x_k$} be a sequence satisfying $x_{k+1}\le(1-\beta)x_k$ for $0\lt\beta\lt 1$ , and $x_0\le C$. ...
3
votes
3answers
40 views

How to prove these two sets are identical?

This is more a question of the methadology one should use to solve these type of questions: Say there is a set $V \subseteq X \subseteq Y$ and $U \subseteq Y$ such that $$X \setminus V = U \cap X $$ ...
1
vote
1answer
56 views

$\mathbb CP^1 \approx S^2$ proof check

I wanted to give a whole proof of this fact as I was not able to find a detailed one myself. I have the feeling that such a proof has been asked quite frequently by several users and I hope this may ...
2
votes
4answers
59 views

Show that $6^n/n! \le 6^5/5! \times 6/n$

I want to show that $$\frac{6^n}{n!} \le \frac{6^5}{5!} \cdot \frac 6n$$ without using induction, which I've done but is rather clunky. Is there a more straight forward way of doing this?
5
votes
5answers
180 views

Looking for a non-combinatorial proof that $a! \cdot b! \mid (a+b)!$

(I use $a$ and $b$ to denote natural numbers.) Question. Without appealing to the combinatorial interpretation of $$\frac{(a+b)!}{a! b!}$$ as a multinomial coefficients, is there a proof that for ...
0
votes
1answer
33 views

Alternative proof of $\prod_{n=1}^{+\infty}\frac{e^{it/2^n}+1}{2}=\frac{e^{it}-1}{it}$

Let $t\in \mathbb{R}$. I want an alternative proof of the following identity $$\prod_{n=1}^{+\infty}\frac{e^{it/2^n}+1}{2}=\frac{e^{it}-1}{it} \quad(\star)$$ I've came up with this identity observing ...
3
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1answer
47 views

Is there any way to gain some insight into a proof by simply looking at a graphic?

My school is using Pinter's "A Book of Abstract Algebra" for both semesters of Modern Algebra. For a class assignment a couple weeks ago, regarding rings, I was tasked with the following problem ...
2
votes
1answer
77 views

Proving that product of transpose matrix and the matrix is inversible

I need to prove that $A^T$$A$ is an invertible matrix. $$ A= \begin{bmatrix} \vec{a_1} & \vec{a_2} & \ldots & \vec{a_n} \\ \end{bmatrix} $$ Can I prove this using ...
2
votes
4answers
83 views

If $ f \rightarrow c$ then prove $\frac{1}{a} \int_{[0,a]} f \rightarrow c$

Let $f$ be an extended real-valued $\mathcal{M}_{L}$-measurable function on $[0,\infty)$ such that $f$ is $\mu_L$-integrable on every finite subinterval of $[0,\infty)$, and $$ \lim_{x\rightarrow ...
1
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2answers
42 views

Simple Inequality of Complex Numbers, $\left| \frac{a-b}{1-\overline{a}b} \right| <1$

Exercise from Ahlfor's Complex: Given $a,b \in \mathbb C$, with $|a| <1$, $|b|<1$, prove: $$\left|\frac{a-b}{1-\overline{a}b}\right| <1.$$ My argument: Lemma: If $\alpha, \beta \in ...
2
votes
1answer
42 views

Show that a piecewise function of two solutions to an ODE is a solution to the ODE

Let $x=x(t)$. If the first order ODE $x'=f(t,x) (*)$ is satisfied by $u$ and $v$, each over $I = (a,b)$, show that $$w(t) := u1_{(a,t_0)} + v1_{[t_0,b)}$$ satisfies $(*)$, where $u(t_0) = v(t_0)$ ...
0
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1answer
20 views

Show code $C$ is a 1 error correcting

Let $C=\{(0,0,0,0,0),(1,1,1,0,0),(0,0,1,1,1),(1,1,0,1,1)\}\in\mathbb{F}_2$. Show $C$ is $1$ error correcting. Definition: a code $C\subseteq\mathbb{F}^n$ is t error correcting , if for any two ...
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2answers
38 views

Proof using pumping lemma that $\{0^m1^n \mid m \neq n \}$ is not regular

First of all, there's already a question very similar to this, but in my case I just wanted to show my attempt with the hope that if there are any erros or it's wrong completely you can help me in the ...
1
vote
1answer
69 views

For all $x$, $f(x)=\int_{0}^{x}f(t)dt$. Prove that $f(x)=0$ for all $x$.

Suppose that the function $f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous and that $$f(x)=\int_{0}^{x}f(t)dt\qquad\text{for all $x$}$$ Prove that $f(x)=0$ for all $x$. Attempt Since ...
4
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0answers
52 views

Monomorphisms of monoids are stable under coproducts

Let $M,N,K$ be three monoids (or even groups, if you like) and let $N \to K$ be an injective homomorphism. Then, the induced morphism $M \sqcup N \to M \sqcup K$ is also injective. This is easy to ...
3
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2answers
68 views

Alternative proof: G group of order $p^2$, p prime $\Rightarrow$ $H$ is normal in $G$.

Let $G$ be a group of order $p^2$ where $p$ is prime. If $H$ is a subgroup of order $p$, show that $H$ is normal in $G$. I would like to prove this with the tools that the book has provided up to ...
0
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1answer
22 views

On a Simple Theorem from Hilbert's *The Foundations of Geometry*

I want my proof writing skills to get better. I am trying to do this through proving theorems from Hilbert's axioms for Euclidean Geometry. I found Hilbert's The Foundations of Geometry here, a ...
1
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2answers
38 views

Is there a name for this double summation identity? What is the shortest way to illustrate that it holds?

Say I have the following the expression: $$\sum\limits_{j=0}^{i-1} \sum\limits_{u=0}^{j} g(u)$$ By enumeration, it is easy to see that: in the case when $j=0$ we have $$\sum\limits_{u=0}^{j} g(u) ...
0
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0answers
33 views

Use Cauchy-Schwarz inequality to prove that $\langle\,,\rangle : \mathscr H \times \mathscr H \to \Bbb C$ is continuous.

Let $(a,b) \in \mathscr H \times \mathscr H$ be fixed. So we have to prove that for a given $\epsilon \gt 0$, we can find $\delta_1 \gt 0$ and $\delta_2 \gt 0$ such that $\lvert \langle x,y\rangle - ...
0
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0answers
46 views

Are 2 quadrilaterals similar if they are both inscribed and have congruent angles and have perp diagonals

This is problem 365 from Kiselev's Planimetry book. I have to show that two inscribed quadrilaterals with perpendicular diagonals are similar iff they have respectively congruent angles. Here is my ...
0
votes
2answers
49 views

Can one show that $\frac{\vec{u}\cdot \vec{v}}{||\vec{u}||*||\vec{v}||}$ is always on the range of $\cos \theta $?

A basic property of the dot product of two vectors is that $$\frac{\vec{u}\cdot \vec{v}}{||\vec{u}||*||\vec{v}||} = \cos \theta $$ Where $\theta$ is the angle between the two vectors. Since there is ...
1
vote
1answer
57 views

Prove $T$ is invertible

If $T\in L(X,X)$ where $X$ is a Banach space and $L(X,X)$ is denoted as the space of bounded linear maps, and $\|I-T\|<1$ where $I$ is the identity operator, then $T$ is invertible? Here ...
0
votes
0answers
23 views

Easy computations using the functional equations for Riemann and Gamma functions

Let $\zeta(z)$ the Riemann Zeta function and $\Gamma(z)$, the Gamma function. I've deduced easily an equation involving these functions. I don't known if it is useful, if there are mistakes in my ...
1
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0answers
39 views

Solving a Diophantine Equation with 2 variables

This is my answer for the following question: Find all natural numbers $(a,b)$ for which $a^b-b^a=1$. When $a$ or $b$ equals $1$, $(a,b)=(2,1)$ is trivial. If $a,b>1$, I generalized the problem ...
1
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0answers
35 views

Direct proof of the existence of optimal memoryless deterministic policies in MDP

It is well known that (finite-state, finite-action, discrete time) MDPs admit an optimal policy that is memoryless and deterministic (sometimes called pure). The proof of this fact for ...