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

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44
votes
12answers
6k 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 ...
34
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 ...
31
votes
3answers
2k 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 ...
30
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 ...
26
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 ...
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. ...
23
votes
3answers
626 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
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 ...
21
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 ...
19
votes
4answers
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 ...
18
votes
3answers
831 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
6k 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 $0$ ...
16
votes
1answer
261 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 ...
15
votes
4answers
3k 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
2answers
335 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 ...
14
votes
1answer
772 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
139 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 ...
13
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 ...
13
votes
2answers
607 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
219 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
438 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 $\,f(x)=f(y),\,$ ...
12
votes
5answers
271 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
261 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
189 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
4answers
6k 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
1answer
233 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 ...
11
votes
1answer
77 views

Valid proof of Young's Inequality?

Part of an exercise to prove Holder's inequality in Rudin involves proving Young's Inequality... That is, given $\frac{1}{p}+\frac{1}{q} = 1$, prove $$ab \leq \frac{a^p}{p} + \frac{b^q}{q}.$$ ...
10
votes
4answers
465 views

How can I complete this proof by contradiction?

This problem is from Discrete Mathematics and its Applications: Prove that there are no solutions in integers $x$ and $y$ to the equation $2x^2 + 5y^2 = 14$. I am trying to use proof by ...
10
votes
2answers
149 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 ...
10
votes
4answers
757 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 ...
10
votes
1answer
96 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 ...
10
votes
2answers
463 views

Why does this cube root trick work?

So, I found that you can work out cube roots between $1$ and $100$ using this method: For example the number: $185193$ If the last digit $x$ is $2$, $8$, $3$, or $7$, the last digit in the answer ...
10
votes
1answer
261 views

Showing a sequence of integrals converges to zero

Let $\varphi$ be a complex-valued function which is analytic on $\{z \in \mathbb C : |z| \leq 2\}$, let $\gamma$ be the unit circle in the complex plane, and define $$ F_n(z) = \int_\gamma ...
10
votes
0answers
238 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 ...
9
votes
5answers
569 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 ...
9
votes
5answers
966 views

Number Theory : Infinitude Of Primes - a different proof

I was doing some basic Number Theory problems and came across this problem : Show that the integer : $Q_{n} = n ! + 1$, where $n$ is a positive integer, has a prime divisor greater than $n$. ...
9
votes
1answer
189 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
733 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. ...
9
votes
2answers
140 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 / ...
9
votes
1answer
254 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) $\ ...
9
votes
1answer
105 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
323 views

Separation in compact spaces

There was recently a question that I cannot find about separation in compact spaces. The answer to that question was no for trivial reasons. Motivated by that, let me ask a less trivial version of ...
9
votes
2answers
689 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 $\tau$ ...
9
votes
0answers
229 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 ...
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
358 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
292 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
270 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
713 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. ...