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
24 answers

Proofs of AM-GM inequality

The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual proofs on the internet but I was wondering if someone…
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$n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem?

If $n\geq2$, then $n!$ is not a perfect square. The proof of this follows easily from Chebyshev's theorem, which states that for any positive integer $n$ there exists a prime strictly between $n$ and $2n-2$. A proof can be found here. Two weeks and…
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What is the most unusual proof you know that $\sqrt{2}$ is irrational?

What is the most unusual proof you know that $\sqrt{2}$ is irrational? Here is my favorite: Theorem: $\sqrt{2}$ is irrational. Proof: $3^2-2\cdot 2^2 = 1$. (That's it) That is a corollary of this result: Theorem: If $n$ is a positive…
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Visual proof of $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$?

In his gorgeous paper "How to compute $\sum \frac{1}{n^2}$ by solving triangles", Mikael Passare offers this idea for proving $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$: Proof of equality of square and curved areas is based on another…
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Is it technically incorrect to write proofs forward?

A question on an assignment was similar to prove: $$2a^2-7ab+2b^2 \geq -3ab.$$ and my proof was: $$2a^2-4ab+2b^2\geq0$$ $$a^2-2ab+b^2\geq0$$ $$(a-b)^2\geq0$$ which is true. However, my professor marked this as incorrect and the "correct" way to do…
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Proving that $\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$

Prove without evaluating the integrals that:$$2\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx=\int_\frac{\pi}{2}^\pi\frac{x\ln(1-\sin x)}{\sin x}dx\label{*}\tag{*}$$ Or equivalently: $$\boxed{\int_0^\pi\frac{x\ln(1-\sin x)}{\sin…
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Surprisingly elementary and direct proofs

What are some examples of theorems, whose first proof was quite hard and sophisticated, perhaps using some other deep theorems of some theory, before years later surprisingly a quite elementary, direct, perhaps even short proof has been found? A…
Martin Brandenburg
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Alternative proofs that $A_5$ is simple

What different ways are there to prove that the group $A_5$ is simple? I've collected these so far: By directly working with the cycles: page 483 of Because it has order 60 and two…
3 answers

A $\{0,1\}$-matrix with positive spectrum must have all eigenvalues equal to $1$

Here's a cute problem that was frequently given by the late Herbert Wilf during his talks. Problem: Let $A$ be an $n \times n$ matrix with entries from $\{0,1\}$ having all positive eigenvalues. Prove that all of the eigenvalues of $A$ are…
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What are the theorems in mathematics which can be proved using completely different ideas?

I would like to know about theorems which can give different proofs using completely different techniques. Motivation: When I read from the book Proof from the Book, I saw there were many many proof for the same theorem using completely…
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The Hexagonal Property of Pascal's Triangle

Any hexagon in Pascal's triangle, whose vertices are 6 binomial coefficients surrounding any entry, has the property that: the product of non-adjacent vertices is constant. the greatest common divisor of non-adjacent vertices is constant. Below is…
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**Ended Competition:** What is the shortest proof of $\exists x \forall y (D(x) \to D(y)) $?

The competition has ended 6 june 2014 22:00 GMT The winner is Bryan Well done ! When I was rereading the proof of the drinkers paradox (see Proof of Drinker paradox I realised that $\exists x \forall y (D(x) \to D(y)) $ is also a theorem. I…
6 answers

New bound for Am-Gm of 2 variables

Today I'm interested by the following problem : Let $x,y>0$ then we have : $$x+y-\sqrt{xy}\leq\exp\Big(\frac{x\ln(x)+y\ln(y)}{x+y}\Big)$$ The equality case comes when $x=y$ My proof uses derivative because for $x\geq y $ the function…
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Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula

As far as every question I've seen concerning "what is $\sum_{k=1}^nk^p$" is always answered with "Faulhaber's formula" and that is just about the only answer. In an attempt to make more interesting answers, I ask that this question concern the…
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The equation $x^3 + y^3 = z^3$ has no integer solutions - A short proof

Can someone provide the proof of the special case of Fermat's Last Theorem for $n=3$, i.e., that $$ x^3 + y^3 = z^3, $$ has no positive integer solutions, as briefly as possible? I have seen some good proofs, but they are quite long (longer than…
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