For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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27 views

A problem on path-wise connectedness

Let $K = \{(x,y) \in \mathbb{R^2}| x=0,-1 \leq y \leq 1\}$, $G=\{(x,y) \in \mathbb{R^2}| \ 0<x \leq 1, y=\text{sin}(\frac{1}{x})\}$ and $A=K\bigcup G.$ Claim: $A$ is not pathwise connected. ...
4
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2answers
207 views

A Householder matrix is symmetric

I want to show that a Householder matrix is symmetric, so I must show that $H^T = H$, but from the formula $$H= I - (uu^T/\beta),$$ they are not equal. What's wrong with my reasoning? EDIT: I ...
7
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4answers
264 views

For which primes p is $p^2 + 2$ also prime?

Origin — Elementary Number Theory — Jones — p35 — Exercise 2.17 — Only for $p = 3$. If $p \neq 3$ then $p = 3q ± 1$ for some integer $q$, so $p^2 + 2 = 9q^2 ± 6q + 3$ is divisible by $3$, ...
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5answers
187 views

Intuition — An integer $n > 1$ is composite $\iff \color{purple}{p \le \sqrt{n}}$ divides it.

Origin — Elementary Number Theory — Jones — p32 — Lemma 2.14 Backward direction — I need to prove there exists a divisor $d$ of $n$ satisfying $1<d<n$. Because $p$ is prime, $1 < p$. ...
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2answers
121 views

Proof - There're infinitely many primes of the form 3k + 2 — origin of $3q_1..q_n + 2$

Origin — Elementary Number Theory — Jones — p28 — Exercise 2.6 To instigate a contradiction, postulate $q_1,q_2,\dots,q_n$ as all the primes $\neq 2 (=$ the only even prime) of the form $3k+2$. ...
4
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1answer
563 views

Proof — Infinitely many primes of the form $4k + 3$ — origin of $4(p_1…p_k - 1) + 3$

I know there are sundry questions — like this pdf — and this (10.) Prove that any positive integer of the form $4k + 3$ must have a prime factor of the same form. Because $4k + 3 = 2(2k + 1) + 1$, ...
2
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4answers
79 views

Proof strategy to show $\large \sum_{0 \ \le \ x \ \le \ a} \normalsize \binom{a}{x} \binom{b}{n+x} = \binom{a+b}{a+n} $

The following two combinatorial identities are taken from a textbook. $$\begin{align} &\large \sum_{0\ \le \ x \ \le \ n} \normalsize \binom{a}{x} \binom{b}{n-x} = \binom{a+b}{n} \tag{10} \\ ...
1
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3answers
89 views

For a finite field $F$ of order $n$, all elements are roots of $x^n - x$

I need to prove two things at $F[X]$ but don't know how, Ill glad to get help... $F$ is a finite field. $|F|=n$. We look at $p(x)=x^n-x\in F[X]$ 1. How we can show that every $c\in F$ is a root of ...
1
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1answer
32 views

Is the approach to proving this expression for an n-choose-k algorithm correct?

I randomly encountered this post here, asking why this is true: ...
5
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3answers
77 views

Do Question's Given GCD Statements Imply these New GCD Statements?

Are the following statements true or false, where $a$ and $b$ are positive integers and $p$ is prime? In each case, give a proof or a counterexample: (b) If $\gcd(a,p^2)=p$ and ...
0
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2answers
84 views

General Proof that $\left< v, v \right> = \left|\left| v \right|\right|^2$

Consider $\mathbb{R}^n$ with the standard Euclidean inner-product. I'm trying to give a proof that $$ \left< v, v \right> = \left|\left| v \right|\right|^2 $$ but can only seem to do it for ...
0
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1answer
39 views

Commutative property based on integral substraction.

I found elementary algebra exercise which I can't resolve. Let algebraic structure $(X,\cdot)$ where $\cdot$ has properties: $$ x\cdot(x\cdot y)=y \\ (y\cdot x)\cdot x=y $$ How to proof that ...
1
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1answer
28 views

Connection between proof and algebraic factor theorem

It is (no proof needed) Let $f \in End(V)$ and $ f^2 = f$ Then $V = Ker(f) \oplus Im(f)$. What does this mean for the following sentence?? "Factor Theorem. Let $F : V \to W $ be linear and $A = ...
3
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0answers
60 views

Proofs involving Disjunctions [Velleman, Chapter 3.5]

$\Large{{1.}}$ Are proofs using strategies $P136, P143$ always easier than those using $P140$? In the former two, only one statement (either $P$ or $Q$) must be proven. In the latter, both $P$ and ...
2
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2answers
41 views

How does the fundamental theorem of algebra extend to show number (in addition to existence) of roots?

The fundamental theorem of algebra in which we prove a complex polynomial has at least root is clear from the construction of a compact domain and use of the polar coordinate form of complex numbers. ...
0
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1answer
26 views

If $(D_n)_{n \in \mathbb{N}}$ is such that $2014 = \frac {2013^{D^{31}_n +1} } {2013^{D^{31}_{n+1}}-2014^{0.1}},$ then $\lim(D_n) < \pm \infty.$

Given a sequence $(D_n)_{n \in \mathbb{N}}$ such that $2014 = \frac {2013^{D^{31}_n +1} } {2013^{D^{31}_{n+1}}-2014^{0.1}}, \forall n\in \mathbb{N}.$ How do I show that $\lim(D_n) < \pm \infty ?$ ...
1
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3answers
109 views

Arithmetic and geometric mean

I need to prove that for $a=\frac{x+y}{2}$ and $g=\sqrt{xy}$, following statments are true or false: For $x\not =y,a>g$ and $x=y, a=g$. I have no idea how to do this, so any help is welcomed. ...
0
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3answers
62 views

Deductions from Carmichael's theorem

1.) May I seek your advice on how to prove the following assertion(without recourse to (2)): If $ 2014 \equiv 14 \ (\text{mod} \ 2000), $ then $2014^{2014} \equiv 14^{14} \ (\text{mod 2000}).$ 2.) ...
5
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1answer
90 views

Rational distance from an equilateral triangle

Is there a nice proof for the following fact? In a plane, there does not exist a square such that its vertices are at a rational distance from each vertex of some equilateral triangle. What if ...
0
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1answer
21 views

seat every club member np problem

A University has n clubs, the largest of which contains m members (students can be members of multiple clubs). The President of the University wishes to hold a dinner in honor of such student ...
2
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5answers
125 views

Proving that if $a^2+b^2=c^2$, then $a+b\ge c$.

Hello, I'm trying to prove this statement. Let a,b & c be three positive real numbers and if $a^2+b^2=c^2$ then $a+b\ge c$ Any help, please?
3
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2answers
94 views

$x$ and $y$ be distinct elements of order $2$ in $G$ that generate $G$. Prove that $G \cong D_{2n}$ [duplicate]

Problem Let $G$ be a finite group and let $x$ and $y$ be distinct elements of order $2$ in $G$ that generate $G$. Prove that $G \cong D_{2n}$, where $\vert xy \vert = n$. Solution We have $x^2 = ...
0
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1answer
65 views

How do i prove $\text{F}_\text{n+1}^2 - F_n\text{F}_\text{n+2} = (-1)^n$ by induction? [duplicate]

$F_n$ refers to the $n$ term of the Fibonacci Sequence. I think i'm suppose to prove this by induction. I already have the base case. I am at: $\text{F}_\text{k+1}^2 - F_k\text{F}_\text{k+2} + ...
2
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3answers
427 views

Prove that the additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic.

Is my proof below correct? What specific property of rationals did I exploit in my proof? It looks like the property I exploited is the following: Given any positive rational, I can always write it as ...
2
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3answers
238 views

Prove that the additive groups $\mathbb{R}$ and $\mathbb{Q}$ are not isomorphic.

Is there a better (or other) way(s) to prove the following statement? Also, the same argument works for multiplicative groups $\mathbb{R}-\{0\}$ and $\mathbb{Q}-\{0\}$, right? Problem Prove that ...
12
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2answers
297 views

Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.

Is my proof correct? I have made use of the fact isomorphism preserves order of elements, which I proved couple of exercises back. I am also interested in other ways of proving it. Is there a more ...
6
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3answers
157 views

Intuition and Tricks - Crafty Short Proof - Generators, Order of a Cyclic Group - Fraleigh p. 64 Theorem 6.14

This stronger result and easier proof is based on p. 58. Hence it isn't a duplicate of this. Theorem 206 and 207. Let $G$ be a group, $k \in \mathbb{N}$ and $a \in G$ such that $|a| = n$. Then: 206. ...
2
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1answer
68 views

Show that $GL_n(\mathbb{F})$ is a finite group iff $\mathbb{F}$ has a finite number of elements

I would like to know if my proof below is correct. Problem Show that $GL_n(\mathbb{F})$ is a finite group iff $\mathbb{F}$ has a finite number of elements. Solution If $\mathbb{F}$ is a ...
4
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3answers
117 views

Proving that $L \le M$ as limits of $f$ and $g$ when $f(x) \le g(x)$

While doing some tasks for my next calculus course, I ran across this task: "Let $a < b <c$, and assume that $f(x) \le g(x)$ for all $x \in [a, c]$. If $\lim_{x \to b}f(x) = L$ and $\lim_{x \to ...
2
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2answers
70 views

If $\omega$ is closed on $\mathbb R ^2 - 0$ and $\text d \omega =0$, then $\omega = \text d g+ \lambda \text d \theta$.

I'm trying to solve problem 4-30 from “Calculus on manifolds”, which is the one in the title, where $$\text d \theta = -\dfrac{y}{x^2+y^2}\text d x+\dfrac{x}{x^2+y^2}\text d y.$$ I think I'm on the ...
5
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3answers
179 views

Intuition and Tricks - Hard Overcomplex Proof - Order of Subgroup of Cyclic Subgroup - Fraleigh p. 64 Theorem 6.14

Update Dec. 28 2013. See a stronger result and easier proof here. I didn't find it until after I posted this. This isn't a duplicate. Proof is based on ProofWiki. But I leave out the redundant $a$. ...
2
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0answers
143 views

Show that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles.

I would like to know if my proof below is correct. Problem Let $p$ be a prime. Show that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles. Show by ...
1
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3answers
91 views

countable or not countable

Good evening everyone; Can you tell me whether are they countable or not ? $$ 2^\mathbb{R}\\ 2^\mathbb{Z}\\ 2^\mathbb{N} $$ where $\mathbb{R}$ is the set of real numbers. $\mathbb{Z}$ is set of ...
0
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1answer
161 views

Mathematical derivation of successive parabolic interpolation

I'm studying for an exam, and in some past papers they ask to provide 'a mathematical derivation of the iteration scheme'. Now, I understand successive parabolic interpolation, I can do it (or I ...
1
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0answers
70 views

Prove that $\sqrt{\frac{2a}{a+b}}+\sqrt{\frac{2b}{b+c}}+\sqrt{\frac{2c}{c+a}}\le3$ [duplicate]

Prove that $$\sqrt{\frac{2a}{a+b}}+\sqrt{\frac{2b}{b+c}}+\sqrt{\frac{2c}{c+a}}\le3$$ where $a$ and $b$ and $c$ are positive real numbers. How to do it? Thank you
1
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1answer
64 views

If $\tau = (1,2)(3,4,5)$ determine whether there is a $n$-cycle $\sigma$ ($n\geq 5$) with $\tau = \sigma^k$ for some $k\in\mathbb N$.

I would like to know if my proof below is correct. Problem If $\tau = (1,2)(3,4,5)$ determine whether there is a $n$-cycle $\sigma$ ($n\geq 5$) with $\tau = \sigma^k$ for some integer $k$. ...
3
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1answer
68 views

Elementary number theory problem

Let $X = \{n \in \mathbb{N}: 6 \times n\,\, \text{does not consist of} \ 0,1,2,3 \, \text{or} \ 4\}.$ For eg, $93 \in X$ because $6 \times 93=558.$ Could anyone advise me how to prove there ...
0
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2answers
157 views

If two polynomials both of n degree have n identical real roots, are they equal? Proof?

CORRECTION: The polynomials don't have to be equal, but one has to be a constant multiple of the other. I ask the question because I saw this fact used in this solution to a problem: Problem: Given ...
5
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1answer
225 views

Tricks to Prove Union of Two Subgroups iff One is Contained in the Other - Fraleigh p. 54 - based on Exercise 5.4.5

Not a duplicate because I'm asking about tricks and the blueprint for the proof based on this. Let $H, K \le G$. Prove $H \cup K$ is a subgroup $\iff H \subseteq K$ or $K \subseteq H$. Backward ...
4
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2answers
276 views

Intuition - An inverse of a generator is a generator and powers of generators - Fraleigh p. 58 5.46

(1). How do you envisage or envision a Cyclic Group with only one generator can have at most 2 elements? Solution is based on this. The integers have two generators, 1 and -1. Which should give you ...
0
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2answers
55 views

continuous map from indiscrete space

Show that a continuous map from indiscrete space X to a $T_0$ space must be constant function. I was trying to prove it by contradiction. If the map f is not constant then there is two points x, y ...
4
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4answers
419 views

If we accept a false statement, can we prove anything? [duplicate]

I think that the question is contained in the title. Suppose we begin from something that is false for example $1=0$. Is it possible using only $\Rightarrow$ (and of course $\lnot ,\wedge,\lor$) to ...
2
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2answers
66 views

combinatorics proof with i - Peter J. Cameron book

I'm working through Peter J. Cameron's combinatorics book and I'm having trouble understanding one of his proofs. In proposition 3.3.3, he states: "If n is a multiple of $8$, then the number of sets ...
0
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1answer
67 views

Finishing proof of identity $\sum_{k=b}^{n} \binom{n}{k} \binom{k}{b} = 2^{n-b} \binom{n}{b}$

The identity $$ \sum_{k=b}^{n} \binom{n}{k} \binom{k}{b} = 2^{n-b} \binom{n}{b}\ $$ is one of a few combinatorial identities I having been trying to prove, and it has taken me way too long. I am ...
0
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5answers
167 views

Questions about Divisibility of $2^n$ by $3$

Why is it that $\forall n \in N$, $2^n$ is not divisible by $3$? I can prove it easily by induction, but I don't understand the intuition of why this is true. Could anyone please supply the ...
6
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2answers
122 views

Intuition/How to determine if onto or 1-1, given composition of g and f is identity. [GChart 3e P239 9.72]

9.72. $A,B$ are nonempty sets. $f: A \rightarrow B$ and $g: B \rightarrow A$ are functions. Suppose $g \circ f = $ the identity function on $A$. (♦) Are the following true or false? $1.$ $f$ ...
5
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2answers
308 views

Proof Strategy - Tricky Question involving distinct, non-zero eigenvalues of $A^{2}$ - 2011 8C

Remember that we've already proven the following, for any real symmetric $n\times n$ matrix $M$: (i) Each eigenvalue of $M$ is real. (ii) Each eigenvector can be chosen to be real. (iii) Eigenvectors ...
3
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3answers
73 views

How to prove that triangle inscribed in another triangle (were both have one shared side) have lower perimeter?

This question looks really simple, but to my (and my co-workers) frustration we were not able to prove this in any way. I tried all triangle formulas known to me but I feel I'm missing the point, and ...
16
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2answers
796 views

Integral $\int_1^\infty\dfrac{dx}{1+2^x+3^x}$

Can the integral $$\int_1^\infty\dfrac{dx}{1+2^x+3^x}$$ be given in closed form? This question arises naturally when I considered doing integrals. What makes an integral hard? Well, the integrand, of ...
12
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4answers
330 views

Tips for writing proofs

When writing formal proofs in abstract and linear algebra, what kind of jargon is useful for conveying solutions effectively to others? In general, how should one go about structuring a formal proof ...