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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3
votes
2answers
71 views

$P(AB=BA)$ , $A,B\in M_{3x3}(\mathbb Z/p\mathbb Z)$

Let $A,B\in M_{3x3}(\mathbb Z/p\mathbb Z)$ ($p$ a prime number). Find the probability $P$ that $AB=BA$ that is $P(AB=BA)$ $$A=\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} ...
2
votes
3answers
84 views

Prove that if A is ANY $n\times n$ matrix, then $det(adj(A)) = (det(A))^{n-1}$. (how to when A is singular?)

Prove that if $A$ is ANY $n\times n$ matrix, then $det(adj(A)) = (det(A))^{n-1}$. first of all, since I did it about 3 or 4 times before, I started off by proving the case where $A$ is an INVERTIBLE ...
0
votes
2answers
36 views

Find integers $r$, $s$, and $t$ such that $12r + 30s + 18t = 2$

Could someone please explain if such integers exist and how to find them? If not, could someone please explain how to prove that they don't exist? Thank you!
4
votes
2answers
159 views

Walk on Earth: Math Puzzle

Here's the famous math puzzle posted by Prof. Walter Lewin about a person walking on earth, quoted below for posterity: A person stands on the North Pole. She walks 10 miles South, then 10 ...
2
votes
2answers
126 views

Prove that if $A$ is a square matrix with integer entries and $\det(A) = \pm 1$, then $A^{-1}$ contains all integer entries.

Prove that if $A$ is a square matrix with integer entries and $\det(A) = \pm 1$, then $A^{-1}$ contains all integer entries. I'm really thrown off by this one, its unlike all the examples I've seen.. ...
0
votes
1answer
39 views

Return to sum of powers question.

Previously I had asked a question about a Diophantine equation linked here. I have come back to think about this question but in a different manner. So here is the set up: Let A and $a_i$ be ...
1
vote
3answers
72 views

Proof by induction of a sum?

Let $n ∈ N$. Prove by induction that there are $n$ ways to write the number $n$ as a sum $n=x_1+x_2+...+x_k$ where the $x_i$ are natural numbers and $x_1 ≤x_2 ≤...≤x_k ≤x_1+1$. For example, $5 = 5$, ...
0
votes
1answer
62 views

Real Analysis Question: derivatives

Let $$f''(x)+p(x)\cdot f(x)=0$$ and $$g''(x)+p(x)\cdot g(x)=0$$ where $a<x<b$. 1 ) Show that $W=f'g-fg'$ is a constant on $(a,b)$. 2 ) Prove: If W$\neq$0 and $f(x_1)=f(x_2)=0$ where $a \lt x_1 ...
2
votes
1answer
53 views

Verification of identity $2\sum_{m,n\geq 1\,;\,m>n}\frac{1}{m^k n^k}=\left(\sum_{n\geq 1}\frac{1}{n^k}\right)^2 -\sum_{n\geq 1}\frac{1}{n^{2k}}$ [closed]

Is this identity true? $$2\sum_{m,n\geq 1\,;\,m>n}\frac{1}{m^k n^k}=\left(\sum_{n\geq 1}\frac{1}{n^k}\right)^2 -\sum_{n\geq 1}\frac{1}{n^{2k}}$$ If so, how to prove it? Could you provide me a ...
7
votes
2answers
150 views

Prove that $\sum_{k=0}^\infty \frac{1}{16^k} \left(\frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 + 712k^2 + 194k + 15}\right) = \pi$

How to prove the following identity $$\sum_{k=0}^\infty \frac{1}{16^k} \left(\frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 + 712k^2 + 194k + 15}\right) = \pi$$ I am totally clueless in this one. Would ...
1
vote
4answers
403 views

Use the division algorithm to prove that 3|(n³ + 2n) for all n ∈ ℕ

I can do it by induction, thanks to the wonderful people of this website, but I'm not sure how to do it by the Division algorithm. Can anyone help me? I think I can show how 3 divides 2n, but I'm not ...
2
votes
2answers
55 views

Show that $\Gamma_f:\mathbb{R}\to\mathbb{R}^2$ by $\Gamma_f(x)=(x,f(x))$ is continuous, with $f$ continuous.

The entire problem statement is, Let $f:\mathbb{R}\to\mathbb{R}$ be continuous. Define $\Gamma_f:\mathbb{R}\to\mathbb{R}^2$ by $\Gamma_f(x)=(x,f(x))$. Show that $\Gamma_f$ is continuous. My attempt ...
0
votes
3answers
87 views

Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for all $s, t \in \mathbb{Z}$

Would this be the same thing as saying "Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for any $s, t \in \mathbb{Z}$"? I can do the proof for any integers $s$ and $t$, but if any and all mean the same ...
2
votes
2answers
67 views

Prove or disprove that if a|(sb+tc) for all (and for some) s,t ∈ ℤ, then a|b, and a|c.

So, this is actually 2 questions in 1. I apologize if that is bad practice, but I didn't want to write 2 questions when they're a word different. So, I have Prove or disprove that if $a|(sb+tc), ...
1
vote
1answer
127 views

Conditional proof/contradiction, long example problem

Here are the premises/conclusion, and where I've gotten so far. $1.$ $(W\wedge E)\rightarrow (P\vee L)$ (PR) $2.$ $(W\wedge \neg E)\wedge R))\rightarrow (P\vee D)$ (PR) $3.$ $((W\wedge \neg ...
1
vote
1answer
54 views

Asymptotic Normality of MLE when data is modelled with covariates

Say I have data vector $X_1,\ldots,X_n$ which I want to model with some parametric distribution function $f(X_i;\theta,Z_i)$ and covariates $Z_i$. In this case, how can I prove the asymptotic ...
0
votes
1answer
58 views

Real Analysis using the Archimedean Principle

Let $x$ be a real number. Then there exists a natural number $n$ such that $3^n > x$. proof: Let $x$ be a real number. Suppose that $x$ is an upperbound of the natural numbers. We know $1$ is a ...
0
votes
2answers
39 views

How do I begin to prove the limit of this definite integral?

I was fooling around with a graphing calculator, and I noticed a pattern in the functions of $$x^a+y^a=1$$ where $a$ is an even number. As $a$ increases, the graph begins to look like a square (if you ...
2
votes
3answers
48 views

Show that $G_{s}$ is a normal subgroup of $G$

Definition: $G_{s}:=\{g \in G: g.s=s\}$ My attempt is the following: We take $g \in G$, and we consider this two sets: $$gG_{s}:=\{gh:h\in G_{s} \}$$ $$G_{s}g:=\{hg :h\in G_{s}\}$$ and we will ...
4
votes
2answers
63 views

How to prove the following bounds expression

Let n be a positive integer. Prove that there are 2^(n−1) ways to write n as a sum of positive integers, where the order of the sum matters. For example, there are 8 ways to write 4 as the sum of ...
0
votes
1answer
25 views

Let $T$ be a mobius transformation such that $T(3i)=5$ and $T$ maps circle $\{|z-i|=4\}$ onto circle $\{|z-2|=2\}.$ Determine all values of $T(9i) $

Let $T$ be a mobius transformation such that $T(3i)=5$ and $T$ maps circle $\{|z-i|=4\}$ onto circle $\{|z-2|=2\}.$ Could anyone advise me how to find all possible values of $T(9i) \ ?$ A mobius ...
0
votes
1answer
25 views

Prove that a $\kappa : G/G_{s} \to G.s$ is a bijection

I have to prove that given an action this function $\kappa : G/G_{s} \to G.s$ is a bijection. $$ G/G_{s} \to G.s$$ $$gG_{s} \to g.s$$ Where $G$ is a group and: $G_{s}:=\{g \in G : g.s=s\}$(Isotropy ...
5
votes
1answer
100 views

Proving $f(x)$ attains $\max$ or $\min$ when $f(x)\to0$ as $|x|\to\infty$.

Suppose $f:\Bbb R\to\Bbb R$ is continuous such that $f(x)\to0$ as $|x|\to\infty$. Prove that $f$ attains either a maximum or a minimum. My attempt at the question : Given $\epsilon > 0 \ \ ...
-1
votes
2answers
58 views

least value of a complex number

If $z_1,z_2,z_3,z_4\in C $ satisfy $z_1+z_2+z_3+z_4=0$ and $|z_1|^2+|z_2|^2+|z_3|^2+|z_4|^2=1$ then what will be the least value of $|z_1-z_2|^2+|z_1-z_4|^2+|z_2-z_3|^2+|z_3-z_4|^2$? What approach ...
3
votes
1answer
95 views

Find a Mobius transformation $f$ that maps $\mathbb{H}=\{z \in \mathbb{C}:Im(z) >0\}$ bijectively to ball $B(0,2)$ such that $f(i)=1, f(1)=-2 \ ?$

Could anyone advise me on this problem: Find a Möbius transformation $f$ that maps $\mathbb{H}=\{z \in \mathbb{C}:\text{Im}(z) >0\}$ bijectively to ball $B(0,2)$ such that $f(i)=1, f(1)=-2 \ ?$ ...
1
vote
2answers
221 views

Proof Question using Proof By Contradiction, irrationality of $a + \sqrt[b]{5}$

I answered this question but am not quite sure if i did what was correct. If anything is wrong please point it out, thanks. Question:Prove that any number of the form $a + \sqrt[b]5$ is irrational, ...
1
vote
3answers
109 views

Confirm definite integral equals zero $\frac{\sin(x)}{(1-a\cos(x))^{2}}$

Is this statement about the definite integral of a particular function $F$ true? $$\int_0^{2\pi}F(x)\, \mathrm{d}x = \int_0^{2\pi}\frac{\sin(x)}{(1-a\cos(x))^2}\, \mathrm{d}x = 0 \ \text{ for }\ ...
0
votes
2answers
110 views

Let A be an invertible nxn matrix. Prove that $\det(\operatorname{adj}(A^{-1})) = (\det(A))^{1-n}$

Let $A$ be an invertible $n\times n$ matrix. Prove that $\det(\operatorname{adj}(A^{-1})) = (\det(A))^{1-n}$ I tried starting with $A^{-1} = 1/\det(A) \cdot \operatorname{adj}(A)$ I tried everything ...
1
vote
0answers
75 views

Any general hints on how to prove that two functions$\ f(n)$ and$\ g(m_1,m_2,…,m_{28})$ never have a common natural divisor?

All the variables are natural numbers. I'm not asking for a proof, since while we simply have$\ f(n)=n^3-n+1$,$\ g$ is a very long sum of cube roots (which contain square roots as well). I'm after ...
0
votes
2answers
65 views

How to prove the following expression

Prove that if it takes you 5 minutes to solve any Sudoku puzzle and 14 minutes to solve a word search, you can completely occupy yourself on any flight of 52 minutes or longer provided that you have a ...
1
vote
0answers
72 views

suggestion on how to learn maths effectively

this question is not about a problem.My problem is I was made to read topics such as real analysis,complex analysis,metric spaces,topology,functional analysis,abstract algebra comprising of group ...
-1
votes
2answers
37 views

Use division algorithm and then induction to show 3|(n³+2n) for all ℕ. [duplicate]

For division algorithm, would I do something along the lines of n³+2n = 3q+r and go from there? For induction, I did the base case, which is true, and so then I moved on to the k+1 case, in which I ...
1
vote
1answer
83 views

Prove that if $p\ge 5$ is prime, then $p^2 + 1$ is composite

So, coming off of this question, I know how to find out what the remainder is, so after figuring whether the remainder is $1$ or $5$, would I just plug in $p = 6q + (1\ \text{or}\ 5)$ into $p^2+1$? ...
1
vote
2answers
440 views

Prove that the equation $x^{3}-3x+b=0$ has at most one root in the interval $[-1,1]$

I have to prove that the equation $x^{3}-3x+b=0$ has at most one root in the interval $[-1,1]$. My attempt: We consider the function $g(x)=x^{3}-3x+b$.Now since it is a polynomial it is ...
2
votes
1answer
72 views

Prove that $p \ge 5$ is prime, then the remainder of $p$ upon division by $6$ is $1$ or $5$.

An example in my textbook, but I'm not quite sure how to set this one up, because of the $p \ge 5$ part. How do I start it off?
1
vote
0answers
32 views

Finding the highest power of N to fit in a given power of 2

I am trying to find the highest power $p$ of a number $N$ that will fit in a given power of 2. To give this some context, I am trying to find the largest power of $N$ that will fit in a 64-bit signed ...
0
votes
2answers
91 views

How to prove that if m and n are natural numbers than m+n is also a natural number?

Problem sounds easy enough - prove that if $m$ is in set of all natural numbers (let's call it $\mathbb N$) and so is $n$ than $m+n$ also must be there. Probably it should be done using induction. But ...
2
votes
3answers
103 views

Prove that $2\sqrt{n}\sqrt{n+1} < 2n + 1$ for all positive integers.

I've been testing this with many values and it seems to always be true. I've been trying to rework the inequality into a form where it's much more obvious that the left hand side is always less than ...
2
votes
3answers
117 views

Rational number arbitrarily close to a square root of 2 [duplicate]

I am trying to prove the proposition by contradiction For all rational $c > 0$, there exists a rational number $x$ such that $x^2 < 2 < (x + c)^2$. with the negation $x^2 \ge 2 \lor 2 ...
0
votes
1answer
98 views

Proof involving the Pigeonhole principle

Prove that among any given $n + 1$ positive integers, there are always two whose difference is divisible by $n$ My Answer: Using Pigeonhole principle: From a set of at least $2$ different $n+1$ ...
2
votes
1answer
74 views

How to prove uniqueness of *wannabe* final object in a slice category?

I am beginning to study category theory, and I think I need your help to find my way in this sea of uncertainty (!). I have the following problem (n. $5.11$ from Aluffi's Algebra: Chapter $0$). Let ...
1
vote
4answers
127 views

A intersection (A union B)

I'm trying to prove $A \cap (A \cup B) = A$. I'm stuck on the last part of my proof, not sure how to show next: $$x \in A \cap (A \cup B)$$ $$\iff x \in A \;\;\text{and}\;\; x \in A \cup B$$ $$\iff x ...
1
vote
3answers
154 views

Proof verification: A $(16,5,8)$ binary code does exist.

Well I have used spheres in coding with radius, $r=\left\lfloor\frac{\delta -1}{2} \right\rfloor=\left\lfloor\frac{8 -1}{2} \right\rfloor = 3$ and that means we have $\sum \limits_{i=0}^3 {16 \choose ...
0
votes
2answers
30 views

Is there some trick to manipulating an equation? (adding 0s, multiplying by 1, etc..)

I have such a hard time doing this sort of thing that it's annoying me. I'm not very mathematically inclined but it frustrates me that a solution with such a small answer takes me more than a page to ...
4
votes
3answers
124 views

Prove the gcd $(4a + b, a + 2b) $ is equal to $1$ or $7$.

So in the question it says to let $a$ and $b$ be nonzero integers such that $\gcd(a,b) = 1$. So based on that I know that $a$ and $b$ are relatively prime and that question is basically asking if the ...
2
votes
3answers
137 views

Counting the number of different ways in which groups of one or two can be formed…

I'm having trouble proving that the number of ways n>3 people can be divided into groups of either one or two is equal to: $A_n = A_{n-1} + (n-1)⋅A_{n-2} $ I'm trying to prove this by counting but ...
1
vote
3answers
78 views

Proof that a sequence is convergent

I'm asked to prove the convergence of the sequence $$X_n=\left(1+\frac12\right)\left(1+\frac14\right)\left(1+\frac18\right)\cdots\left(1+\frac{1}{2^n}\right)$$ I proved that it is increasing through ...
1
vote
1answer
263 views

The Brownian motion process in Sheldon M. Ross

Today I study Brownian Motion and Geometric Brownian Motion using textbook: An Elementary Introduction to Mathematical Finance, Third Edition by Sheldon M. Ross but I missed the class because I was ...
3
votes
1answer
96 views

Prove $\forall n \in N$, every set of natural numbers of size n has a maximum element. May assume that sets do not repeat numbers.

Prove using induction. So i'm a bit confused about how to do this question. My attempt at it seems like i'm missing a lot and it looked to easy. ...
1
vote
1answer
30 views

For integer $n$ prove that if there is no integer $m\le \sqrt{n}$ such that $ m | n$, then $n$ is prime.

At first I thought that the best way to prove this statement is to take the direct approach and show the subset {1, 2, 3,...sqrt(n)} and the subset {sqrt(n),... n/3, ..., n/2,...,n} and show that ...