Questions tagged [alternative-proof]

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

3172 questions
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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 of $M$ such that $\phi(M)\subseteq\mathfrak{a}M$.…
33
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Is there an idempotent element in a finite semigroup?

Let $(G,\cdot)$ be a non-empty finite semigroup. Is there any $a\in G$ such that: $$a^2=a$$ It seems to be true in view of theorem 2.2.1 page 97 of this book (I'm not sure). But is there an elementary proof? Theorem 2.2.1. [R. Ellis] Let $S$ be a…
32
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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 here) but I had to use the L'Hospital's Rule…
Paramanand Singh
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A trivial proof of Bertrand's postulate

Write the integers from any $n$ through $0$ descending in a column, where $n \geq 2$, and begin a second column with the value $2n$. For each entry after that, if the two numbers on that line share a factor, copy the the entry unchanged, but if…
30
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Proof $\int_{-\infty}^\infty\frac{dx}{\left(e^x+e^{-x}+e^{ix\sqrt{3}}\right)^2}=\frac{1}{3}$

I'm interested in possible generalizations of the integral $$ \int_{-\infty}^\infty\frac{dx}{\left(\cosh x+\frac{1}{2}e^{ix\sqrt{3}}\right)^2}=\frac{4}{3},\tag{1} $$ or equivalently $$ \int_0^\infty\frac{dt}{(1+t+t^{\,\alpha})^2}=\frac23, \quad…
27
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6 answers

Why Markov matrices always have 1 as an eigenvalue

Also called stochastic matrix. Let $A=[a_{ij}]$ - matrix over $\mathbb{R}$ $0\le a_{ij} \le 1 \forall i,j$ $\sum_{j}a_{ij}=1 \forall i$ i.e the sum along each column of $A$ is 1. I want to show $A$ has an eigenvalue of 1. The way I've seen…
27
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1 answer

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 out to be periodic with period 9, i.e., $$ a_n =…
26
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4 answers

How to prove this algebraic version of the sine law?

How to solve the following problem from Hall and Knight's Higher Algebra? Suppose that \begin{align} a&=zb+yc,\tag{1}\\ b&=xc+za,\tag{2}\\ c&=ya+xb.\tag{3} \end{align} Prove that …
25
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Prove that $2018^{2019}> 2019^{2018}$ without induction, without Newton's binomial formula and without Calculus.

Prove that $2018^{2019}> 2019^{2018}$ without induction, without Newton's binomial formula and without Calculus. This inequality is equivalent to $$ 2018^{1/2018}>2019^{1/2019} $$ One of my 'High school' student asked why the inequality is true. The…
Elias Costa
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Integral $\int_0^\infty \frac{\ln x}{(\pi^2+\ln^2 x)(1+x)^2} \frac{dx}{\sqrt x}$

I have stumbled upon the following integral:$$I=\int_0^\infty \frac{\ln x}{(\pi^2+\ln^2 x)(1+x)^2} \frac{dx}{\sqrt x}=-\frac{\pi}{24}$$ Although I could solve it, I am not quite comfortable with the way I did it. But first I will show the way. We…
Zacky
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A Ramanujan sum involving $\sinh$

Today, in a personal communication, I was asked to prove the classical result $$\boxed{ \sum_{n\geq 1}\frac{(-1)^{n+1}}{n^3\sinh(\pi n)} = \frac{\pi^3}{360}}\tag{CR} $$ which I believe is due to Ramanujan. My proof can be found here and it is based…
25
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A proof of Wolstenholme's theorem

This was inspired by this question. I tried to use the identity $${2n \choose n}=\sum_{k=0}^n {n \choose k}^2$$ (see this question) to prove that $$\binom{2p}p\equiv2\pmod{p^3}$$ if $p\gt3$ is prime. (Wikipedia gives a combinatorial proof that this…
joriki
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An elementary proof of $\int_{0}^{1}\frac{\arctan x}{\sqrt{x(1-x^2)}}\,dx = \frac{1}{32}\sqrt{2\pi}\,\Gamma\left(\tfrac{1}{4}\right)^2$

When playing with the complete elliptic integral of the first kind and its Fourier-Legendre expansion, I discovered that a consequence of $\sum_{n\geq 0}\binom{2n}{n}^2\frac{1}{16^n(4n+1)}=\frac{1}{16\pi^2}\,\Gamma\left(\frac{1}{4}\right)^4 $…
23
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Prove that if $A$ is normal, then eigenvectors corresponding to distinct eigenvalues are necessarily orthogonal (alternative proof)

The problem statement is as follows: Prove that for a normal matrix $A$, eigenvectors corresponding to different eigenvalues are necessarily orthogonal. I can certainly prove that this is the case, using the spectral theorem. The gist of my proof…
Ben Grossmann
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Alternative and more direct proof that an integral is independent of a parameter

I'm looking for alternative ways to calculate the integral $$ \int\limits_0^\infty\frac{\tanh(\alpha x)}{\tanh(\pi x)}\sin(2\alpha x^2)\,dx=\frac{1}{4},\qquad \alpha>0.\tag {*} $$ It was derived in a lengthy calculation using roundabout method from…