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

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26
votes
3answers
1k views

Fascinating Lampshade Geometry

Today, I encountered a rather fascinating problem in a waiting room, which is embodied in the image below. Notice how the light is being cast on the wall? There is a curve that defines the ...
23
votes
2answers
1k 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 ...
21
votes
5answers
3k 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 ...
17
votes
3answers
712 views

My first induction proof (very simple number theory). 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 ...
16
votes
10answers
1k views

Assume that the square matrix A has an eigenvalue of 0. Is A invertible? Why or why not?

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 and invertible and for the sake of contradiction let $0$ ...
15
votes
5answers
1k 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 ...
13
votes
4answers
2k 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 ...
13
votes
2answers
216 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 ...
12
votes
3answers
1k 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
1answer
210 views

Teaser or fun calc equation to surprise husband (physicist/EE) at work

I am a geneticist and unfortunately have not worked much with advanced calc since undergrad. In genetics, as you likely know, a male is denoted as XY and a female as XX. I plan to leave a riddle for ...
12
votes
2answers
299 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 ...
11
votes
5answers
363 views

Where's the problem with a false “proof”: $\;1^0 = 1^2 \overset{?}\implies 0 = 2$

What's wrong with this: $$\large 1^0=1^2$$ Since bases are same, therefore $$\large 0=2$$ My thinking: Since the function $\,f(x)=1^x\,$ is not one to one, therefore whenever $\,f(x)=f(y),\,$ ...
11
votes
4answers
685 views

Scratch work for delta-epsilon proof for $\lim_{x \to 13} \sqrt{x-4} = 3$

Prove $\lim_{x \to 13} \sqrt{x-4} = 3$. We need to show for all $E> 0$ there exists $D > 0$ such that if $0 < |x - 13| < D$, then $|\sqrt{x-4} - 3| < E$. Let me write D for ...
11
votes
0answers
77 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 $e'$ ...
10
votes
4answers
2k 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 ...
9
votes
1answer
160 views

Is my proof correct for: $\sqrt[7]{7!} < \sqrt[8]{8!}$

I have to show that $$\sqrt[7]{7!} < \sqrt[8]{8!}$$ and I did the following steps \begin{align} \sqrt[7]{7!} &< \sqrt[8]{8!} \\ (7!)^{(1/7)} &< (8!)^{(1/8)} \\ (7!)^{(1/7)} - ...
9
votes
1answer
99 views

Proving $(2n-1)^n + (2n)^n ≈ (2n+1)^n$

As I do, I was messing around and I thought to myself this simple thing: $3^2 + 4^2 = 5^2$ I just thought that this is only Pythagorean triplet with sequential integers. I know that there are no ...
8
votes
7answers
2k views

Prove that $4$ is the only solution to $2+2$. [duplicate]

This question was featured on Saturday Morning Breakfast Cereal and I haven't been able to find a proof. Can anyone help?
8
votes
4answers
260 views

Any group of order $85$ is cyclic.

Any group of order $85$ is cyclic. My attempt: Let $|G|=85=5\times17$ Let $H_1$ be the sylow-$5$ and $H_2$ be sylow-$17$ subgroup of $G.$ Then $H_1\cap H_2=\{e\}$ So $|H_1\times H_2|=85$ and ...
8
votes
3answers
168 views

Polynomial irreducible - maximal ideal

I have a couple of ideals which I wonder if I correctly classify as maximal/prime ideal. $I_1 = \langle 2x^2 + 9x -3\rangle$, $I_2 = \langle x - 1\rangle$ $\mathbf 1)$ Is $I_1$ a maximal ideal in ...
8
votes
2answers
89 views

Proof of non-existence of non-principal $\kappa$-complete ultrafilter

Let $\lambda$ be a cardinal. I would like to prove that for all cardinals $\lambda < \kappa \leq 2^\lambda$, there can't be a $\kappa$-complete non-principal ultrafilter on $\kappa$. Here is my ...
8
votes
1answer
137 views

Validity of my weird proof that $AB$ and $BA$ have the same eigenvalues?

On a recent linear algebra exam, I was required to prove that "for every $n \times k$ matrix $A$ and $k \times n$ matrix $B$ over the same field, it holds that $AB$ and $BA$ have the same eigenvalues ...
8
votes
1answer
96 views

Proof by induction that $\alpha^n + \beta^n \in \mathbb Z$

Let $\alpha, \beta \in \mathbb C$ such $\alpha + \beta \in \mathbb Z$ and $\alpha \beta = j \in \mathbb Z$. Prove that for all $n \in \mathbb N,\alpha^n + \beta^n \in \mathbb Z$ Is this proof ...
8
votes
1answer
193 views

Question on Riemann sums

Question is : What I have read in Riemann sum definition is something like $$\sum_{i=1}^n f(y) .|x_i-x_{i-1}|$$ So, at first sight i am afraid this is not even related to Riemann integration of ...
8
votes
1answer
168 views

My proof of the First Fundamental Theorem of Calculus

I've tried to prove the theorem in advance at the level that satisfies me. The notation used might not be correct, but I hope all major steps are correct. DEFINITION (from Apostol's Calculus I): Let ...
7
votes
3answers
812 views

Bad proof or not?

Can you prove this: Let $a,b \in \mathbb{N}$. If $a + b + ab = 2020$ then $a+b=88$. This is the attempt given: $\frac{2020-88}{a b}=1$ $a+b=88$ Substituting for b using the 2nd equation. ...
7
votes
5answers
355 views

If $\mathcal{P}(A)=\mathcal{P}(B)$, then $A=B$? [duplicate]

Prove, disprove, or give a counterexample: If $\mathcal{P}(A)=\mathcal{P}(B)$, then $A=B$. Assume $\mathcal{P}(A)=\mathcal{P}(B)$. Since we know $A \subseteq A$, we know $A \in ...
7
votes
4answers
93 views

Herstein Question: $G^{i}$ normal in $G$?

I just wanted to ask a quick question. I'm going over the second edition of I.N. Herstein's topics in algebra and one of his exercises asks the reader to prove that each $G^{i} $ is a normal subgroup ...
7
votes
1answer
301 views

If $\phi: G \mapsto H$ is an isomorphism, prove that $G$ is abelian if and only if $H$ is abelian

I would like to know if my proof is correct. Specifically, I would like you to check that surjectivity is needed for proving the first part and injectivity is needed for proving the second part. ...
7
votes
2answers
114 views

Does the $\gcd(2n-1,2n+1)=1?$

I am posting this to ask if my proof is correct as I haven't taken number theory in a year and I feel a bit rusty. If it isn't correct, please tell me where I went wrong so I can fix it. I want to ...
7
votes
2answers
118 views

Which is correct? Group Theory

Unfortunately I noticed that all are wrong. (A) Counterexample: $G=(1)$ (B), (C), (D) Counterexample: $G=\{1,-1\}$ Please help! Where did I go wrong?
7
votes
2answers
326 views

Feedback on my proof that $(A\setminus B)\cup(B\setminus A)=(A\cup B)\setminus (A\cap B)$

I would like to prove that $(A\setminus B)\cup(B\setminus A)=(A\cup B)\setminus (A\cap B)$. Please could you offer some feedback? Firstly, I will show that: $(A\setminus B)\cup(B\setminus ...
7
votes
1answer
114 views

Is $\Bbb Q/\Bbb Z$ artinian as a $\Bbb Z$-module?

I'm confused. Is $\Bbb Q/\Bbb Z$ artinian as a $\Bbb Z$-module? We know that $\Bbb Z_{p^{\infty}} \subset \Bbb Q/\Bbb Z$ is artinian. The following argument is true or not ? $\mathbb Q / ...
7
votes
1answer
205 views

Proof that Fibonacci Sequence modulo m is periodic?

It's well known that the Fibonacci sequence modulo m (where m is any integer) is periodic. I have figured out a proof for this, but upon googling, I found proofs online that were far more complicated. ...
7
votes
1answer
147 views

Counterexample to “Measurable in each variable separately implies measurable”

Some fellow classmates are preparing for a qualifying exam on real analysis, and asked me for help on the following question: Let $ \ f:[0,1]^2\longrightarrow\mathbb{R}$ be such that: (i) $\ ...
7
votes
1answer
132 views

Prove that $ \sum_{1 \le t \le n, \ (t, n) = 1} t = \dfrac {n\phi(n)}{2} $

Problem: Prove that the sum of all integers $ t \in \{ 1, 2, \cdots, n \} $ and $ (t, n) = 1 $ is $ \dfrac {1}{2} n \phi (n) $, where $ \phi $ is the Euler Totient Function. My proof: Define the ...
7
votes
1answer
60 views

The Limit: $\lim_{x \to \infty}\frac{e^{f(x+a)}}{e^{f(x)}}$

I'm doing some challenge review problems and I was wondering whether this proof looked correct: Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function with $\lim_{x \to ...
7
votes
1answer
92 views

Prove $\frac {1}{\cos 0^\circ \cdot \cos 1^\circ} + \ldots +\frac {1}{\cos 88^\circ \cdot \cos 89^\circ}= \frac{\cos 1^\circ}{\sin 1^\circ}$

Prove the following identity: $$\frac {1}{\cos 0^{\circ} \cdot \cos 1^{\circ}} + \ldots +\frac {1}{\cos 88^{\circ} \cdot \cos 89^{\circ}} = \frac{\cos 1^{\circ}}{\sin 1^{\circ}}$$ After hours of ...
7
votes
2answers
206 views

Does this Fermat's Last Theorem's “Proof” employ circular reasoning? [closed]

I happened to find this link which contains a "Simple Proof of Fermat's Last Theorem": http://www.oakton.edu/user/4/pboisver/fermat.html I don't believe in the "proof" because we can just change ...
7
votes
2answers
140 views

Let $G$ be a finite group and $\phi:G \to K$ be a surjective homomorphism and $n \in \mathbb{N}. $ If $K$ has an element of order $n,$ so does $G.$

Let $G$ be a finite group$\ ,\phi:G \to K$ be a surjective homomorphism and $n \in \mathbb{N}. $ If $K$ has an element of order $n,$ so does $G.$ May I know if my proof is correct? Thank you. ...
7
votes
3answers
383 views

Prime number theorem, proof error

Can someone help me find where I made an error in this attempted proof And from there, give me some advice on what I can do to fix it $$M(x)=\sum_{n\leq x}\mu(n)$$ $$\psi(x)=\sum_{n\leq ...
7
votes
2answers
245 views

If $\gcd(m,n)=1$, then $\mathbb{Z}_n \times \mathbb{Z}_m$ is cyclic. [duplicate]

If $\gcd(m,n)=1$, then $\mathbb{Z}_n \times \mathbb{Z}_m$ is a cyclic group. Let's denote $\mathbb{Z}_n=\langle1_n \rangle$ and $\mathbb{Z}_m=\langle1_m \rangle.$ My proof goes as follows: since ...
7
votes
1answer
145 views

Strategy of a purely algebraic proof of Cayley-Hamilton Theorem

Let $p(\lambda)=det(A-\lambda l)$ be the characteristic polynomial of a $n \times n$ matrix $A$. Then $p(A)=O.$ Let $p(\lambda)=p_{0}+p_{1}\lambda+\ldots+p_{n-1}\lambda^{n-1}+p_{n}\lambda^{n}$. ...
7
votes
1answer
143 views

Let $f$ be an entire function which takes every value no more than three times. What can it be?

Let $f$ be an entire function which takes every value no more than three times. What can it be? Consider the singularity at infinity. If it is removable then $f$ is constant. If it is a pole ...
7
votes
1answer
120 views

Proof correctness problem

I was watching this talk by Vladimir Voevodsky which was given at the Institute of Advanced Study in 2006. In his talk the first slide he shows has the following written on it: ...
7
votes
0answers
59 views

Find all subgroups of $\mathbb{Z_2} \times \mathbb{Z_2} \times \mathbb{Z_4}$ isomorphic to the Klein $4$-group - Fraleigh p. 110 Exercise 11.12

I tried to fill in the steps but I'm still confounded by this solution. $|\mathbb{Z_2} \times \mathbb{Z_2} \times \mathbb{Z_4}| = 2\cdot2\cdot4$. $order(\mathbb{Z_2} \times \mathbb{Z_2} \times ...
7
votes
0answers
150 views

Fixed point of a shrinking map. Proof.

Could you please verify my proof? Let $f:X \to X$ be a shrinking map on a compact metric space $(X,d)$. In other words: $d(f(x),f(y)) < d(x,y)$ for all $x,y \in X$. Prove: $f$ has a unique ...
7
votes
0answers
171 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 ...
6
votes
7answers
718 views

Evaluation of $\lim\limits_{n\to\infty} (\sqrt{n^2 + n} - \sqrt[3]{n^3 + n^2}) $

Could you, please, check if I solved it right. \begin{align*} \lim_{n \rightarrow \infty} (\sqrt{n^2 + n} - \sqrt[3]{n^3 + n^2}) &= \lim_{n \rightarrow \infty} \sqrt{n^2(1 + \frac1n)} - ...
6
votes
5answers
610 views

Simpler proof - Non atomic measures

Suppose that $(X,\mathcal{E},\mu)$ is a non-atomic finite measure space (i.e. for every $E \in \mathcal{E}$ with $\mu(E)>0$ there exists $F \subset E$ measurable such that $0<\mu(F) ...