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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1
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1answer
35 views

One more question about the prime factors of binomial coefficient…

Could someone explain me one more thing: why if $p$ is greater than $4n/5$, but less or equal to $n$, then $p$ does not divide $\binom{4n}{3n}$? thank you in advance!
0
votes
1answer
48 views

How can you prove the triangle sum theorem?

Some things to consider: -This theorem has proved very, very many theorems many of which with trig, so you can't use any theorems that have been proven with the triangle sum theorem. -The Triangle ...
3
votes
1answer
32 views

Doubly stochastic matrix proof

A transition matrix $P$ is said to be doubly stochastic if the sum over each column equals one, that is $\sum_i P_{ij}=1\space\forall i$. If such a chain is irreducible and aperiodic and ...
3
votes
1answer
63 views

A few questions about Andy Loo's proof of existence of primes between 3n and 4n…

I have a few questions about Andy Loo's proof (get it here): why, for example, if $2n<p\le3n$, then $p$ does not divide $\binom{4n}{3n}$? Same situation for $\frac{4n}{3}<p\le\frac{3n}{2}$... ...
4
votes
2answers
47 views

A positive integer is equal to the sum of digits of a multiple of itself.

Let $n$ be a positive integer, prove there is a positive integer $k$ so that $n$ is equal to the sum of digits of $nk$. I'm not really sure how I should approach this problem, I tried to do a ...
1
vote
0answers
26 views

How to prove that sums of even powers is divisible by p

For $n\leq (p-2)$ I want to prove that $\sum_{k=0}^{p-1} (r+k)^{n} \equiv 0 \pmod{p}$ It is easy to see that it is true for odd n, since $(-a)^k \equiv -a^k$, and you can just pair up terms since ...
5
votes
1answer
230 views

Conjectured compositeness tests for $N=k\cdot 2^n \pm 1$ and $N=k\cdot 2^n \pm 3$

How to prove these conjectures ? Definition : $\text{Let}~ P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)~ , \text{where}~ m ~\text{and}~ x ~\text{are ...
0
votes
2answers
84 views

Stuck on this proof that $ord(f) = ord(g)$

Let $f, g: \mathbb R \to \mathbb R$ be smooth maps such that $f(a) = g(a') = 0$ and let $\tau, \sigma : \mathbb R \to \mathbb R$ be diffeomorphisms such that $$ \tau \circ f = g \circ \sigma$$ ...
2
votes
3answers
67 views

How to prove $A=(A\setminus B)\cup (A\cap B)$ [duplicate]

How to prove $A=(A\setminus B)\cup (A\cap B)$. I have seen this problem and the solution is clear to me. Initially I was satisfied by my prove but now I think it is wrong. How I have proved ...
2
votes
6answers
1k views

How to prove this approximation for a logarithm? [closed]

I need to prove this approximation, but I am unable to conclude $$\log \left(1+\frac{1}{n}\right) \approx \frac{1}{n}$$
1
vote
0answers
51 views

Proving that the g.c.d of non-consectuive Fibonnaci numbers is also a Fibonacci number

I'm trying to prove that: for non-consecutive Fibonacci numbers, and I know that consecutive Fibonacci numbers are co prime, but I just don't how to prove this using what I know. **EDIT: Lulu has ...
7
votes
4answers
212 views

Looking for a direct proof of the following exercise

A friend of mine told me about the following problem: Let $\{r_n\}$ be a sequence of rational numbers such that $\lim_{n\to\infty}r_n=x\in\Bbb R,$ $r_n\neq x,$ for every $n\in\Bbb N$ and ...
1
vote
2answers
64 views

Try to prove a generalization of the IVT?

Here is the statement : "Let $f: (a,b) \to \mathbb{R}$ a continuous function on $(a,b)$ with $a<b$ and $a,b \in \bar{\mathbb{R}}$. Then for all $u \in \left(\lim \limits_{x\to a} f(x), \lim ...
-1
votes
1answer
100 views

Proof of why there are infinite solutions of $i^x=x$

I recently created the problem $i^x=x$ and was enlightened to the fact that there were infinite solutions. Now I want to know why. Thank you for any help.
11
votes
5answers
112 views

Proving the integral series $\int _0^1\left(1-x^2\right)^n\,dx=\frac{2}{3}\cdot \frac{4}{5}\cdot\ldots\cdot \frac{2n}{2n+1}$

We have the series $\left(I_n\right)_{n\ge 1\:}$ where $$I_n=\int _0^1\left(1-x^2\right)^n\,dx.$$ Prove that $$I_n=\frac{2}{3}\cdot \frac{4}{5}\cdot\ldots\cdot \frac{2n}{2n+1}.$$ I tried to ...
2
votes
0answers
34 views

What Are the Indeterminates in the Polynomials in This Proof?

To prove that every field $F$ has an algebraic closure, the first step is to prove that there exists an extension $F_1$ such that each polynomial in $F[x]$ has a root in $F_1$. Whereas in the ...
0
votes
1answer
33 views

If $ax + by = 8$, what is $\operatorname{gcd}(a, b)$?

Our instructor has given us this problem: If $ax + by = 8$, what is $\operatorname{gcd}(a, b)$? I'm confused. Is it not just $8$? Since, say, $\operatorname{gcd}(a, b) = n$, so there must exist ...
-1
votes
3answers
84 views

How to prove that this series converges? [closed]

Prove that $$ \sum_{n=2}^{\infty}\frac{\log (n+1)-\log n}{(\log n)^2}$$ converges.
2
votes
1answer
28 views

If $P^r$ has all positive entries, then so does $P^n$

Let $P$ be the transition probability matrix of a Markov Chain. Argue that it for some positive integer r, $P^r$ has all positive entries, then so does $P^n$, for all integers $n\geq r$ I ...
-1
votes
2answers
67 views

Is there a rigourous proof about this :$(C)'=0$ , where C is any constant?

Is there someone who can give me a rigourous proof about the derivative of any constant $C$ is zero :$(C)'=0$ Note :I know only this for any real number $x>0$:$(C=Cx^0)'=C'=0Cx^{0-0}=0.1=0$ ...
4
votes
3answers
81 views

Proof of $\sum^{2N}_{n=1} \frac{(-1)^{n-1}}{n} = \sum^{N}_{n=1} \frac{1}{N+n}$

The title pretty much summarizes my question. I am trying to prove the following: $$\displaystyle \forall N \in \mathbb{N}: \sum^{2N}_{n=1} \frac{(-1)^{n-1}}{n} = \sum^{N}_{n=1} \frac{1}{N+n}.$$ I ...
0
votes
3answers
64 views

The sequences $\{f_n\},\{f_n'\}$ both are uniformly convergent on $[0,1]$.

Let $f_n(x) = \frac{\ln(1 + n^2x^2)}{n^2}, x \in [0,1]$. Then the sequences $\{f_n\},\{f_n'\}$ both are uniformly convergent on $[0,1]$. Here, $f_n'(x) = \frac{2x}{1 + n^2x^2}$. Both limits of ...
4
votes
3answers
153 views

Accumulation points of sets

Determine all of the accumulation points of the following sets in $\mathbb{R}^1$ and decide whether the sets are open or closed or neither. I have two problems with the following problems first ...
4
votes
0answers
50 views

Real Induction Over Multiple Variables?

I've seen in several different places* that one can use normal mathematical induction to prove the truth of a statement that relies not on just one variable (say, $x$,) but multiple variables (for ...
4
votes
3answers
59 views

Induction proofs for subsets of integers

I know that induction can be used to prove that certain results hold true for all integers, all positive integers, all negative integers, all rational numbers and so on. What I'm noticing from listing ...
2
votes
2answers
35 views

Proofs: Induction on Handsakes

Here is the problem: Suppose $n$ people are at a party, and some number of them shake hands. At the end of the party, each guest $G_i$, $1 \leq i \leq n$ shares that they shook hands $x_i$ times. ...
2
votes
2answers
23 views

The simplest way to find a parametrization of the plane projective curve $XZ-Y^2=0$.

I have to explain to some first year math students that the projective algebraic set $\textbf{Z}(XZ-Y^2)\subset\mathbb P^2_k$ is $$V=\{(a^2_0:a_0a_1:a^2_1)\subset\mathbb P^2_k \,:\, \textrm{for}\; ...
6
votes
1answer
106 views

How would I complete my proof that $\int_a^bf(g(x))\,dx = \int_{g(a)}^{g(b)}f(x)\frac{d}{dx}(g^{-1}(x))\,dx$?

Around two years ago during a second semester calculus class, my professor remarked that $\int\sin(x^2)\,dx$ could not be integrated. Being a bit defiant, I tried (in vain) to prove him wrong. While ...
2
votes
0answers
28 views

The proof of the integral test using the contradiction method.

I am currently writing a short note about the proof techniques. I found a random theorem and wanted to write a proof by contradiction as an example. The theorem says The integral ...
1
vote
3answers
82 views

How to prove $\lim_{n \to \infty} \big(\text{sin}(\frac{1}{n+1})+…+\text{sin}(\frac{1}{2n}) \big)$ exists?

Could anyone advise me on how to prove $\lim_{n \to \infty} \big(\text{sin}(\frac{1}{n+1})+...+\text{sin}(\frac{1}{2n}) \big)$ exists ? My attempt: Since $\ a_n = \ n\text{sin}(\frac{1}{2n}) ...
0
votes
1answer
47 views

strings over a 3-letter alphabet

Given a string made up of only letters $x$, $y$ and $z$, we need to prove if it can be changed into a string such that each 3 letter substring of the string is either $xyz$, $yzx$ or $zxy$. Now, the ...
0
votes
0answers
46 views

How do I solve Exercise 6.2.4 (a) of 'How to Prove It' by Velleman?

I spent 6 hours on it, and I couldn't wrap my head around it. The problem is described below. I am stuck on Case 2. 6.2.4. (a) Suppose R is a relation on A, and ∀x∈A∀y∈A(xRy ∨ yRx). (Note that this ...
1
vote
2answers
40 views

Proof the statement

Given a finite aperiodic irreducible Markov Chain, prove that for some $n$ all terms of $P^n$ are positive. I'm little lost in how to prove that, but I know that: $i)$ If a Markov Chain is ...
2
votes
1answer
16 views

Stochastic matrix proof

Every stochastic $n\times n$ matrix corresponds to a Markov chain for which it is the one-step transition matrix. However, not every stochastic matrix $n\times n$ is the two-step transition ...
1
vote
6answers
93 views

In a field $F=\{0,1,x\}$, how does $1 + 1 = x$?

I understand that in a field with two elements $1 + 1 = 0$, but in a field with three I do not understand how $1 + 1 =x$. The work I have done so far is: \begin{align} 1 + 1 &= \{ 0 , 1 , x\}\\ 1 ...
2
votes
2answers
29 views

Greatest common divisors equal?

Let $a,b$ be natural numbers. Show that $gcd(a^n,b^n)$ = ($gcd(a,b)^n)$ for any integer $n$. How I started was first proof by contradiction, and then tried to do an inductive proof when that didn't ...
0
votes
1answer
54 views

Prove that if a|b and b|a, then a=b or a=-b

I'm having difficulty proving that if a|b and b|a, then a=b or a=-b. Logically, it makes sense to me, but I don't know how to express it.
0
votes
2answers
43 views

Is my procedure correct about sequences?

Let $\alpha\in(0,2)$, and the sequence $$x_{n+1}=\alpha x_n +(1+\alpha)x_{n-1} \quad \forall n\geq 1$$ Find the limit in terms of $\alpha$, $x_0$ and $x_1$. Check my work. If $\alpha=1$, ...
2
votes
0answers
60 views

Euler-Mascheroni constant [strategic proof]

I know two proof about the approximation of Euler-Mascheroni constant $\gamma$, but very technical. So I would like to know if someone has a strategic proof to show that $0,5<\gamma< 0,6.$ ...
0
votes
1answer
35 views

Prove that $\{a_n\}$ is bounded [duplicate]

Let $\{a_n\}$ be a sequence an $L$ a real number such that $\lim_{n\to\infty} a_n = L$ Prove that $\{a_n\}$ is bounded This reminds me of the bounded monotone convergence theorem (BMCT) but in ...
-3
votes
1answer
50 views

Prove that $f(x)$ is continuous at $a$

Prove that $f(x)$ is continuous at $a$ $\iff$ for all sequences sequence $\{a_n\}$ with $\lim_{n\to\infty} a_n = a$, $\lim_{n\to\infty} f(a_n) = f(a)$ I have no idea how to start this. Should I use ...
0
votes
1answer
19 views

Proving a Partial Derivative Equivalence Using Taylor Series Expansion?

I'm studying computer vision, and one of the problems in my book is to prove that $\partial f/ \partial x = f(x+1) - f(x)$ It's been a while since I've touched Taylor Series, and so I'm not sure of ...
3
votes
2answers
64 views

If $\{a_n\}$ converges to $A$, then $\{(a_1\cdots a_n)^{1/n}\}$ converges to $A$

Prove that this sequence converges. I can't do it. Let $\{a_n\}$ be a sequence of positive real numbers that converges to a number $A$. Prove that $\{(a_1\cdots a_n)^{1/n}\}$ converges to $A$.
2
votes
2answers
94 views

Is my proof correct about sequences?

Suppose that $\{ a_n\}_n$ is a sequence of real numbers such that $$ (a_{n+1}-a_n) \rightarrow a, \text{ if } \ n \rightarrow \infty. $$ Prove that $$ \frac{a_n}{n} \rightarrow a \, \text{ if } \ ...
0
votes
1answer
75 views

Are $x,y$ rational if $x+y$ is rational and $x-y$ is rational? [closed]

Are $x,y$ rational if $x+y$ is rational and $x-y$ is rational? This question was given in maths class, and I don't know where to start. I would be happy if the answer was included in the proof.
0
votes
3answers
37 views

Given two specific sets, show that one is a subset of another

Given $$X = \{x : x = 4^n-3n-1 ; n\in\mathbb{N}\}$$ and $$Y = \{y : y = 9(n-1); n\in\mathbb{N}\}$$ Prove that $X \subset Y$. I've been struggling with this problem for hours but I couldn't find a ...
2
votes
2answers
65 views

Prove that $2^{9693}-1$ divisible by $7$

Prove that $2^{9693}-1$ divisible by $7$, by more than one way. my try... that, the power divisible by $3$ so it's divisible by $7$ like $2^3,2^6,2^{12}$ and I think it's wrong.
2
votes
2answers
41 views

Probability of drawing white ball after transferring to new urn n times

I am in a probability theory course and could not find the solution to this question anywhere. The assignment is already turned in, and I am asking this for my knowledge and for others who are also ...
0
votes
1answer
66 views

Two statements R and S are logically equivalent iff R $\iff$ S is a tautology.

How do prove the following statement: "Two statements R and S are logically equivalent iff R↔S is a tautology. without using a true table.Would I have to use cases? So far I have done so far is that ...
-1
votes
1answer
64 views

Prove that: $\overbrace{222…222}$(repeated $1980$ times), divisible by $1982$ [closed]

Prove that: $\overbrace{222...222}$(repeated $1980$ times), divisible by $1982$