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

# Questions tagged [alternative-proof]

3172 questions

**35**

votes

**1**answer

### 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 $\phi(M)\subseteq\mathfrak{a}M$.…

user158047

**33**

votes

**7**answers

### 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…

user59671

**32**

votes

**7**answers

### 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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**32**

votes

**3**answers

### 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…

Trevor

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**30**

votes

**1**answer

### 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…

Nemo

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**27**

votes

**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…

user68654

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**27**

votes

**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 =…

David Zhang

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**26**

votes

**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
…

Ramen Nii-chan

- 1,674
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**25**

votes

**15**answers

### 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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**25**

votes

**4**answers

### 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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**25**

votes

**3**answers

### 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…

Jack D'Aurizio

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**25**

votes

**1**answer

### 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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**24**

votes

**1**answer

### 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 $…

Jack D'Aurizio

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**23**

votes

**7**answers

### 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

- 203,051
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**23**

votes

**1**answer

### 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…

Nemo

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