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Here is a funny exercise $$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$ (If you prove it don't publish it here please). Do you have similar examples?

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closed as primarily opinion-based by Najib Idrissi, user98602, user147263, apnorton, user2345215 Jan 31 '15 at 18:16

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ This is related. $\endgroup$ – J. M. is a poor mathematician Nov 3 '10 at 22:35
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    $\begingroup$ I noticed that your expression can be also written as $\sin(x - y) \sin(x + y) = (\cos y + \cos x) (\cos y - \cos x) $ $\endgroup$ – Quixotic Nov 4 '10 at 11:09
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    $\begingroup$ I have tripped up many calculus students with this one: $log(1+2+3)=log1+log2+log3$. I am evil... $\endgroup$ – user641 Dec 8 '12 at 1:23
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    $\begingroup$ @SteveD If only we could find an odd example... $\endgroup$ – peoplepower Jan 13 '13 at 0:31
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    $\begingroup$ Almost an identity: $$\sqrt{123456790}\approx 11111.11111\,.$$ $\endgroup$ – Jakob Werner Jul 12 '14 at 18:47

63 Answers 63

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Let $\sigma(n)$ denote the sum of the divisors of $n$.

If $$p=1+\sigma(k),$$ then $$p^a=1+\sigma(kp^{a-1})$$ where $a,k$ are positive integers and $p$ is a prime such that $p\not\mid k$.

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If we define $P$ as the infinite lower triangular matrix where $P_{i,j} = \binom{i}{j}$ (we can call it the Pascal Matrix), then $$P^k_{i,j} = \binom{i}{j}k^{i-j}$$

where $P^k_{i,j}$ is the element of $P^k$ in the position $i,j.$

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$$ \int_{-\infty}^{\infty}{\sin\left(x\right) \over x}\,{\rm d}x = \pi\int_{-1}^{1}\delta\left(k\right)\,{\rm d}k $$

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