# Tagged Questions

If you already have a proof for some result, but want to ask for a different proof (using different methods).

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### Is there an easy way to see that this simple recurrence is 9-periodic?

In a colloquium talk yesterday, Robert Bryant pointed out that for all initial values $a_0, a_1 \in \mathbb{R}$, the sequence generated by the recurrence relation $$a_{n+1} = |a_n| - a_{n-1}$$ turns ...
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### Does Nakayama Lemma imply Cayley-Hamilton Theorem?

Consider the Cayley-Hamilton Theorem in the following form: CH: Let $A$ be a commutative ring, $\mathfrak{a}$ an ideal of $A$, $M$ a finitely generated $A$-module, $\phi$ an $A$-module endomorphism ...
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### A limit problem $\lim\limits_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$

This is a problem from "A Course of Pure Mathematics" by G H Hardy. Find the limit $$\lim_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$$ I had solved it long back (solution presented in my blog ...
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### Can $\displaystyle\lim_{h\to 0}\frac{b^h - 1}{h}$ be solved without circular reasoning?

In many places I have read that $$\lim_{h\to 0}\frac{b^h - 1}{h}$$ is by definition $\ln(b)$. Does that mean that this is unsolvable without using that fact or a related/derived one? I can of course ...
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### Alternative proof of simple integral inequality

Problem. Let $f\in C^1(\mathbb R)$ such that $f(0) = 0$ and $0 < f'(x) \le 1$. Prove that for all $x\ge 0$ $$\int_0^x f^3(t)\,dt \le \left(\int_0^x f(t)\,dt\right)^{\!\!2}.$$ Below is my ...
I have found the following formula: $$\frac{\operatorname d^n}{\operatorname dx^n}\left(\frac{x}{e^x-1}\right)=(-1)^n\,\frac{n\sum\limits_{k=0}^{n}e^{kx}\sum\limits_{i=0}^{k}(-1)^i\binom{n+1}{i}(k-i)^... 1answer 961 views ### Hahn-Banach theorem: 2 versions I have a question regarding the Hahn-Banach Theorem. Let the analytical version be defined as: Let E be a vector space, p: E \rightarrow \mathbb{R} be a sublinear function and F be a subspace of ... 5answers 327 views ### Checking that a 3-D diagram is commutative When proving certain results I need to use commutative diagrams, some of which quite complicated. My question is: Do we need to check every small square all the time to make sure that they are all ... 1answer 320 views ### Seeking a more direct proof for: m+n\mid f(m)+f(n)\implies m-n\mid f(m)-f(n) If f:\mathbb N\to\mathbb Z satisfies:$$\forall n,m\in\mathbb N\,, n+m\mid f(n)+f(m)$$How to show that this implies:$$\forall n,m\in\mathbb N,\,n-m\mid f(n)-f(m)?$$I was almost incidentally ... 1answer 3k views ### Solving SAT by converting to disjunctive normal form The first well-known NPC problem is the Boolean Satisfiability Problem, which has a proof of being NPC done by Cook (Cook-Levin Theorem). The problem can easily be described the following way: ... 2answers 1k views ### Algebraic proof of a trig matrix identity? I'll put the question first, and then the background, because I'm not sure that the background is necessary to answer the question: I have a geometric proof, but is there an elegant algebraic proof ... 4answers 2k views ### How can I complete this proof by contradiction? This problem is from Discrete Mathematics and its Applications: Prove that there are no solutions in integers x and y to the equation 2x^2 + 5y^2 = 14. I am trying to use proof by ... 1answer 584 views ### The ring of integers of a number field is finitely generated. For a number field K, we define the ring of integers of K to be$$\mathcal{O}_K:=\{x\in K\big|\ (\exists f\in\mathbb{Z}[X])(f\ \text{ is monic and } f(x)=0)\}.$$Is there any easy way to see from ... 1answer 217 views ### Does the functional equation p(x^2)=p(x)p(x+1) have a combinatorial interpretation? A recent question asked about polynomial solutions to the functional equation p(x^2)=p(x)p(x+1). Subsequently, Robert Israel posted an answer showing that solutions are necessarily of the form p(x)=... 12answers 2k views ### How to prove that \lim\limits_{x\to0}\frac{\tan x}x=1? How to prove that$$\lim\limits_{x\to0}\frac{\tan x}x=1?$$I'm looking for a method besides L'Hospital's rule. 4answers 234 views ### Inequality \binom{2n}{n}\leq 4^n I would like to prove the following inequality, for n=0,1,2,...,$$ \binom{2n}{n}\leq 4^n.$$I already proved it by induction, and I'm looking for another proof. 1answer 215 views ### Is my proof correct for: \sqrt[7]{7!} < \sqrt[8]{8!} I have to show that$$\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)} - (...
The following came up when I worked on the answer for a different question (though it was ultimately not used in this form): Proposition. Given positive angles $\alpha,\beta,\gamma,\delta$ with \$\...