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1d
comment There are 6 people. I have to pick 3 teams with two members..how many selections are possible
Line up the people from left to right in order of student number. The leftmost person can choose her partner in $5$ ways. For each of these ways, the leftmost person not yet chosen can choose her partner in $3$ ways, and now it's over. So there are 5\cdot 3$ ways.
1d
comment $r$-cycle to a power $k$ is also an $r$-cycle if and only if $\gcd(k, r) = 1$
For the harder part, suppose $\gcd(k,r)=1$. Then (Bezout) there are integers $x$ and $y$ such that $kx+ry=1$. We can take $x\ge 0$. Then $\sigma=\sigma^1=\sigma^{kx+ry}=(\sigma^k)^x(\sigma^r)^y=(\sigma^k)^x$. So the powers of $\sigma^k$ are the same as the powers of $\sigma$.
1d
reviewed Leave Open Open mathematical questions for which we really, really have no idea what the answer is
1d
reviewed Leave Open Is there something special about 2015?
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reviewed Leave Open Calculus question - Need a push in the right direction
1d
reviewed Leave Open Solving an exponential equation
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reviewed Leave Open Spectrum of $\mathcal{O}(U)$
1d
reviewed Reopen Exercise 3.4 in Rotman's An Introduction to Algebraic Topology
1d
comment Given automatic equation solvers exist, should one know how to solve equations by hand?
A gripe connected with the question: Because of the availability of sophisticated statistical software, computations can be readily done by people who don't necessarily know what they are doing. Once upon a time, the person in the next door office was an elderly statistician who computed on a long obsolete huge noisy machine that had roughly the capabilities of a grocery store calculator, but was much slower. He had to think before calculating.
1d
revised Finding a good comparison/bound for determining the convergence of a series
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1d
revised Finding a good comparison/bound for determining the convergence of a series
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1d
comment Prove that a recurrence relation (containing two recurrences) equals a given closed-form formula.
Note that $3\cdot 2^k$ is not $6^k$.
1d
revised Finding a good comparison/bound for determining the convergence of a series
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1d
answered Finding a good comparison/bound for determining the convergence of a series
1d
comment If $a^2 + p^2 = b^2$ then $2(a+p+1)$ is a perfect square
We can also argue directly from the equation, but I think of the representation theorem as so standard that it is not worthwhile to do so.
1d
comment If $a^2 + p^2 = b^2$ then $2(a+p+1)$ is a perfect square
Suppose $a^2+b^2=c^2$. There are integers $s,t,k$ (with $s$ and $t$ relatively prime and of opposite parity) such that $a=2kst$, $b=k(s^2-t^2)$ and $c=k(s^2+t^2)$. In our case the $k$ is irrelevant.
1d
comment If $a^2 + p^2 = b^2$ then $2(a+p+1)$ is a perfect square
It follows from the usual representation theorem for Pythagorean triples. Not elegant, but it does the job.
1d
comment Can Number Theory be visualized?
One large area of Number Theory is Geometry of Numbers. We have provided only one link, but if you search you will get many hits. And there is geometric intuition behind quite a few proofs, particularly estimates.
1d
revised Convergence of $\int_0^{\infty} x \cos (x^6)\,dx$
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1d
answered Volumes of revolution?