For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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47
votes
15answers
15k views

Zero divided by zero must be equal to zero

What is wrong with the following argument (if you don't involve ring theory)? Proposition 1: $\frac{0}{0} = 0$ Proof: Suppose that $\frac{0}{0}$ is not equal to $0$ $\frac{0}{0}$ is not equal to $0 ...
46
votes
12answers
7k views

Proof that every number ≥ $8$ can be represented by a sum of fives and threes.

Can you check if my proof is right? Theorem. $\forall x\geq8, x$ can be represented by $5a + 3b$ where $a,b \in \mathbb{N}$. Base case(s): $x=8 = 3\cdot1 + 5\cdot1 \quad \checkmark\\ x=9 = 3\cdot3 ...
41
votes
12answers
5k views

Why do I get one extra wrong solution?

I'm trying to solve this equation: $$2-x=-\sqrt{x}$$ Multiply by (-1): $$\sqrt{x}=x-2$$ power of 2: $$x=\left(x-2\right)^2$$ then: $$x^2-5x+4=0$$ and that means: $$x=1, x=4$$ But $1$ is not a ...
38
votes
9answers
1k views

A game with $\delta$, $\epsilon$ and uniform continuity.

UPDATE: Bounty awarded, but it is still shady about what f) is. In Makarov's Selected Problems in Real Analysis there's this challenging problem: Describe the set of functions $f: \mathbb R ...
37
votes
15answers
4k views

Is it a new type of induction? (Infinitesimal induction) Is this even true?

Suppose we want to prove Euler's Formula with induction for all positive real numbers. At first this seems baffling, but an idea struck my mind today. Prove: $$e^{ix}=\cos x+i\sin x \ \ \ ...
35
votes
7answers
2k views

Find the value of $\sqrt{10\sqrt{10\sqrt{10…}}}$

I found a question that asked to find the limiting value of $$10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$$If you make the substitution $x=10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$ it ...
34
votes
3answers
2k views

Fascinating Lampshade Geometry

Today, I encountered a rather fascinating problem in a waiting room: Notice how the light is being cast on the wall? There is a curve that defines the boundary between light and shadow. In my ...
33
votes
1answer
2k views

Checking a Proof of a Theorem

Theorem 1.2 of Bennett and Skinner (Canad. J. Math., 2004) asserts that the Diophantine equation $x^{p} - 4y^{p} = z^{2}$ is unsolvable for every prime $p \geq 7.$ The following is a possible proof ...
30
votes
11answers
1k views

Is this $\gcd(0, 0) = 0$ a wrong belief in mathematics or it is true by convention?

I'm sorry to ask this question but it is important for me to know more about number theory. I'm confused how $0$ is not divided by itself and in Wolfram Alpha $\gcd(0, 0) = 0$ . My question here is: ...
27
votes
5answers
5k views

Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$

I've just started to learn about the tensor product and I want to show: $$(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}.$$ Can you tell ...
27
votes
2answers
2k views

Lamport claims there is an error in Kelley's proof of the Schroeder-Bernstein theorem. What is it?

In section 4.1 of his note How to write a proof, Leslie Lamport mentions an error in Kelley's exposition of the Schroeder-Bernstein theorem: Some twenty years ago, I decided to write a proof of ...
26
votes
10answers
16k views

Is a matrix $A$ with an eigenvalue of $0$ invertible?

Just wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof Suppose $A$ is square matrix and invertible and, for the sake of ...
25
votes
10answers
4k views

Proving $\sqrt 3$ is irrational.

There is a very simple proof by means of divisibility that $\sqrt 2$ is irrational. I have to prove that $\sqrt 3$ is irrational too, as a homework. I have done it as follows, ad absurdum: Suppose ...
25
votes
3answers
3k views

Can't find mistake in an easy proof.

Consider the following theorem. $\textbf{Theorem:}$ for any sets $A, B, C, D$, if $A \times B \subseteq C \times D$ then $A \subseteq C$ and $B \subseteq D$. Then the following proof is given. ...
24
votes
7answers
3k views

What is the flaw of this proof (largest integer)?

Let $n$ be the largest positive integer. Since $n ≥ 1$, multiplying both sides by $n$ implies that $n^2 ≥ n$. But since $n$ is the biggest positive integer, it is also true that $n^2 ≤ n$. It follows ...
23
votes
2answers
544 views

Would this solution of the limit of the sequence be correct?

Let's suppose that I have the sequence $a_n = \frac{1}{n^2} + \frac{2}{n^2} + \frac{3}{n^2} + \ldots + \frac{n}{n^2}, n \in \mathbb{N}$. And I have to find the limit of the sequence as $n \rightarrow ...
23
votes
1answer
2k views

Is this proof of the infinitude of primes valid?

The current issue (May 2015) of the American Mathematical Monthly has a one-line proof that there are an infinite number of primes, and I don't see why it is correct. Here is the proof: If the set ...
22
votes
3answers
673 views

The series $\sum_{n=1}^\infty\frac1n$ diverges

We all know that the following harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ diverges and grows very slowly!! I have seen many proofs of the result but ...
22
votes
1answer
500 views

Proof of $\zeta(2)=\frac{\pi^2}{6}$

While messing around with some integrals, I have found the following proof for $\zeta(2)=\frac{\pi^2}{6}$, but I'm not sure if it is valid: We take a look at the integral $I=\int_0^{\frac{\pi}{2}} ...
21
votes
5answers
4k views

Is a brute force method considered a proof?

Say we have some finite set, and some theory about a set, say "All elements of the finite set $X$ satisfy condition $Y$". If we let a computer check every single member of $X$ and conclude that the ...
18
votes
8answers
9k views

Prove that zero multiplied by zero is equal to zero.

This is my proof: So, $0\cdot0=0$ And we know that $a-a=0$ By substitution, We have $(a-a)(a-a)=0$ Then by simplifying, $a^2-a^2+a^2-a^2=0$ and the we have $0-0=0$, Therefore, $0=0$. I am ...
18
votes
5answers
903 views

Computing $\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, dx$

For $a\ge 0$ let's define $$I(a)=\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right)dx.$$ Find explicit formula for $I(a)$. My attempt: Let $$\begin{align*} f_n(x) &= \frac{\ln\left(1-2 ...
18
votes
3answers
997 views

Proof by induction that $n^3 + (n + 1)^3 + (n + 2)^3$ is a multiple of $9$. Please mark/grade.

What do you think about my first induction proof? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof First, introducing a predicate ...
18
votes
2answers
493 views

Question from Putnam '89: Primes of the form $101\ldots01$

I'm not a math major, but would like to compete in the Putnam. As suggested in other questions here, I'm working some old contest problems. I'd like some input on this attempted proof--general input ...
18
votes
6answers
688 views

Proving that $\arctan(x)+\arctan(1/x)=\pm \pi/2$, could this line of reasoning possibly be correct?

I know that two questions have already been asked about this exercise, but what I'm asking here is if this solution, which sounds rather strange to me, could possibly be correct. The problems is as ...
18
votes
1answer
320 views

Solution of 2nd order linear ODE with regular singular points, and complex exponents at singularity

The steady state temperature distribution of a rod given by: \begin{equation} \frac{\textrm{d}p(x)y'}{\textrm{d}x} - y = 0,\; 0 \leq x \leq 1,\; \text{and} \;y(0) = 0, \end{equation} ...
18
votes
0answers
270 views

Two interview questions

I recently came across two interview questions for admission in B.Math at an university. I gave the two questions a try and want to know if my solutions are correct or not. Q1: Given that ...
16
votes
4answers
3k views

Proof: 1007 can not be written as the sum of two primes.

The claim is: 1007 can be written as the sum of two primes. We want to prove or disprove it. Edit: My professor provided this definition in his previous assignment: An integer $n \geq 2$ is ...
16
votes
1answer
309 views

What is the geometry behind $\frac{\tan 10^\circ}{\tan 20^\circ}=\frac{\tan 30^\circ}{\tan 50^\circ}$?

This identity is solvable by the help of trigonometry identities, but I guess there is an interesting and simple geometry interpretation behind this identity and I can't find it. I found it when ...
16
votes
1answer
400 views

Flaw or not flaw in Excel's RNG?

I have a question about my understanding of an article of B.D. McCullough (2008) about Excel's implementation of the Wichmann-Hill random number generator (1982). First, a bit of context The ...
16
votes
1answer
201 views

There's no cardinal $\kappa$ such that $2^\kappa = \aleph_0$

I am trying to prove that there is no cardinal $\kappa$ such that $2^\kappa = \aleph_0$ . My attempt: We suppose it exists. Since $\kappa<2^\kappa$, in particular, $\kappa<\aleph_0$. But ...
15
votes
1answer
814 views

Alternative way to do this indefinite integral?

Problem : Solve $\displaystyle \int \frac{e^{3x}+1}{e^x+1}dx$. Attempt : Since we have \begin{align} \int\frac{e^{3x}+1}{e^{x}+1}dx & =\int\frac{e^{3x}}{e^{x}+1}dx+\int\frac{1}{e^{x}+1}dx ...
15
votes
4answers
1k views

How many solutions for this equation?

$$ \frac{x-4}{(x-1)} = \frac{1-4}{(x-1)} $$ Can someone tell me how many solutions are there for the above equation? MY APPROACH: I cross multiplied the equations and re-arranged to get a quadratic ...
15
votes
5answers
4k views

Proving that there are infinitely many primes with remainder of 2 when divided by 3

I need to prove that there are infinitely many primes with remainder of 2 when divided by 3. I started out similarly to Euclid's classic proof of an infinite number of prime numbers: Suppose there is ...
15
votes
3answers
2k views

No simple group of order $300$

So I've been trying to prove that there's no simple group of order $300$. This is what I did and I was wondering if it was enough. $|G|=2^2 \cdot 3 \cdot 5^2$. Suppose $G$ is simple. Then there ...
15
votes
5answers
305 views

How to prove $n < \left(1+\frac{1}{\sqrt{n}}\right)^n$

I want to know how to prove the following inequality. For $n = 1, 2, 3, \ldots $ $$ n < \left(1+\frac{1}{\sqrt{n}} \right)^n $$ I tried with math induction but I failed.
15
votes
2answers
1k 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 ...
14
votes
6answers
2k views

Where is the flaw in my proof that the union of two continuous functions is continuous?

Problem. Let $f:A\to\mathbb{R}$ be a continuous function on $A$ and $g:B\to\mathbb{R}$ be a continuous function on $B$ such that $A\cap B=\emptyset$. Let $h:A\cup B\to\mathbb{R}$ be defined by, ...
14
votes
1answer
817 views

A Weaker Version of the ABC Conjecture

The ABC conjecture states that there are a finite number of integer triples (a,b,c) such that $\frac {\log \left( c \right)}{\log \left( \text{rad} \left( abc \right) \right)}>1+\varepsilon $, ...
14
votes
3answers
459 views

If $p$ is a prime and $p \mid ab$, then $p \mid a$ or $p \mid b$.

The proof is already given in the textbook but I tried other way around. Proof by contradiction: Let's assume that $p$ doesn't divide $a$ and $p$ doesn't divide $b$, but $p$ divides $ab$. So ...
14
votes
3answers
187 views

Given $2015$ points, show that it is possible to separate them such that $1007$ of them lie inside the circle

Suppose there are $2015$ points on a plane, no three collinear. Show that you can draw a circle such that exactly $1007$ lie inside the circle. Here is my solution: First draw a line that ...
14
votes
1answer
545 views

My first proof that uses the well-ordering principle (very simple number theory). Please mark/grade.

What do you think about my first proof that uses the well-ordering principle? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof ...
14
votes
0answers
348 views

Show that in any group of order $23 \cdot 24$, the $23$-Sylow subgroup is normal.

Show that in any group of order $23 \cdot 24$, the $23$-Sylow subgroup is normal. Let $P_k$ denote the $k$-Sylow subgroup and let $n_3$ denote the number of conjugates of $P_k$. $n_2 \equiv 1 ...
13
votes
9answers
3k views

Has anyone heard of this maths formula and where can I find the proof to check my proof is correct? $\sum^n_{i = 1}i + \sum^{n-1}_{i=1}i = n^2$

The formula basically is: The sum of all integers before and including $n$, plus all the integers up to and including $n-1$. This will find $n^2$. $$ \sum^n_{i = 1}i + \sum^{n-1}_{i=1}i = n^2 $$
13
votes
3answers
3k views

Showing a compact metric space has a countable dense subset

I know that this is separable, but that has not been covered by the text at this point. I believe my proof (below) to be correct, but not very rigorous. Assuming that it actually is correct, could ...
13
votes
5answers
12k views

Prove that $\sqrt 5$ is irrational

I have to prove that $\sqrt 5$ is irrational. Proceeding as in the proof of $\sqrt 2$, let us assume that $\sqrt 5$ is rational. This means for some distinct integers $p$ and $q$ having no common ...
13
votes
2answers
236 views

Let $f$ and $g$ be two group homomorphisms from $G$ to $G'$. Is $H$ a subgroup of $G$?

Can someone please verify this? Let $f$ and $g$ be two group homomorphisms from $G$ to $G'$. Let $H \subset G$ be the subset $\{x \in G: f(x)=g(x)\}$. Is $H$ a subgroup of $G$? Let $e$ and $e'$ ...
13
votes
0answers
115 views

Function defined as a limit

Q. If $f(x)=\lim\limits_{n\to\infty}\dfrac{\log(2+x)-x^{2n}\sin(x)}{1+x^{2n}}$, then explain why the function does not vanish anywhere in the interval $[0,\pi/2]$, although $f(0)$ and $f(\pi/2)$ ...
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) ...
12
votes
3answers
189 views

Evaluate $\int_0^1\int_0^1 \left\{ \frac{e^x}{e^y} \right\}dxdy$

I want compute this integral $$\int_0^1\int_0^1 \left\{ \frac{e^x}{e^y} \right\}dxdy, $$ where $ \left\{ x \right\} $ is the fractional part function. Following PROBLEMA 171, Prueba de a), last ...