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

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28
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 ...
28
votes
1answer
1k views

Checking a possible proof of Fermat's Last 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 ...
27
votes
7answers
1k 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 ...
25
votes
3answers
2k 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. ...
25
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 ...
22
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
3answers
786 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 ...
17
votes
5answers
2k 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 ...
16
votes
10answers
3k 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 $0$ ...
14
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 ...
14
votes
1answer
743 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
2answers
306 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 ...
13
votes
2answers
495 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 ...
13
votes
2answers
168 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 $e'$ ...
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
5answers
259 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.
12
votes
1answer
239 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
3answers
154 views

Prove that $\frac{(p^{n}-1)(p^{n}-p)…(p^{n}-p^{n-1})}{n!} \in \mathbb{N}$ with $p$ a prime number and $n \in \mathbb{N}$

Apparently this question requires a method linked with linear algebra but I was wondering if it was possible to solve it in a formal way like an induction on $n$ or by using an identity for $p^{n}-1$ ...
11
votes
5answers
427 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
4k 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 ...
11
votes
4answers
729 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
1answer
216 views

Let $f$ be a cont. on $\mathbb{R}$ and define $G(x)=\int_0^{\sin (x)}f(t) dt $. Show that $G$ is differentiable on $\Bbb{R}$ and compute $G'$.

Let $f$ be a continuous function on $\mathbb{R}$ and define $$G(x)=\int_0^{\sin (x)}f(t) dt $$ Show that $G$ is differentiable on $\mathbb{R}$ and compute $G'$. This is an exercise from ...
10
votes
1answer
78 views

There is no continuous mapping from $L^1([0,1])$ onto $L^\infty([0,1])$

There is no continuous mapping from $L^1([0,1])$ onto $L^\infty([0,1])$. Proof: suppose $T:L^1 \rightarrow L^\infty$ continuous and onto. $L^1$ is separable, let $\{f_n\}$ be a countable dense ...
9
votes
1answer
173 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
103 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 ...
9
votes
1answer
469 views

Intersection of topologies

Is my proof that the intersection of any family of topologies on a set $X$ is a topology on $X$ correct? Proof. We are required to show that the intersection satisfies the topology axioms. Let $\tau$ ...
8
votes
5answers
479 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 ...
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
2answers
350 views

Equality of two iterated square roots

Solve for $x$: $\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\dots}}}}=\sqrt{x\sqrt{x\sqrt{x\sqrt{x\dots}}}}$ My attempt: The L.H.S is equal to $\dfrac{1+\sqrt{4x+1}}{2}$ and R.H.S equals $x^2$ Equating both ...
8
votes
4answers
278 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
225 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
1answer
702 views

Cannot find a mistake in an incorrect proof.

I am reading an introductory book on proof-writing techniques. One of the exercises asks to demonstrate why the proof is incorrect. I spent quite a while thinking about it and still feel puzzled. ...
8
votes
1answer
564 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. ...
8
votes
2answers
130 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 / ...
8
votes
1answer
209 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) $\ ...
8
votes
1answer
353 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. ...
8
votes
2answers
97 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
44 views

Every representation of a finite group is reducible?

I somehow "proved" that every representation of a finite group is reducible. While I'm fairly sure the error is something silly, I can't seem to place it. Could someone please help me figure out what ...
8
votes
1answer
177 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
101 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
211 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
247 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
4answers
853 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
819 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) ...
7
votes
4answers
107 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
2answers
124 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
375 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
2answers
119 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
3answers
83 views

valid proof of series $\sum \limits_{v=1}^n v$

$$\sum \limits_{v=1}^n v=\frac{n^2+n}{2}$$ please don't downvote if this proof is stupid, it is my first proof, and i am only in grade 5, so i haven't a teacher for any of this 'big sums' proof: if ...
7
votes
1answer
151 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 ...