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1500 questions
124
votes
15 answers
Infiniteness of non-twin primes.
Well, we all know the twin prime conjecture. There are infinitely many primes $p$, such that $p+2$ is also prime. Well, I actually got asked in a discrete mathematics course, to prove that there are infinitely many primes $p$ such that $p + 2$ is…
Tomas Wolf
- 1,381
124
votes
8 answers
Probability that a stick randomly broken in five places can form a tetrahedron
Edit (June. 2015) This question has been moved to MathOverflow, where a recent write-up finds a similar approximation as leonbloy's post below; see here . Randomly break a stick in five places. Question: What is the probability that the resulting…
Benjamin Dickman
- 16,198
124
votes
1 answer
Lebesgue measure theory vs differential forms?
I am currently reading various differential geometry books. From what I understand differential forms allow us to generalize calculus to manifolds and thus perform integration on manifolds. I gather that it is, in general, completely distinct from…
sonicboom
- 10,353
124
votes
1 answer
Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?
A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$ . Is there a characterization of the groups in which this property holds? If this question is too broad,…
Alexander Gruber
- 28,325
124
votes
7 answers
What are the issues in modern set theory?
This is spurred by the comments to my answer here . I'm unfamiliar with set theory beyond Cohen's proof of the independence of the continuum hypothesis from ZFC. In particular, I haven't witnessed any real interaction between set-theoretic issues…
Paul VanKoughnett
- 5,662
124
votes
11 answers
Is zero odd or even?
Some books say that even numbers start from $2$ but if you consider the number line concept, I think zero ( $0$ ) should be even because it is in between $-1$ and $+1$ (i.e in between two odd numbers). What is the real answer?
mvar2011
- 1,391
124
votes
4 answers
Why do we define quotient groups for normal subgroups only?
Let $G \in \mathbf{Grp}$ , $H \leq G$ , $G/H := \lbrace gH: g \in G \rbrace$ . We can then introduce group operation on $G/H$ as $(xH)*(yH) := (x*y)H$ , so that $G/H$ becomes a quotient group when $H$ is a normal subgroup. But why do we only work…
Aleksei Averchenko
- 7,697
124
votes
20 answers
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 integer and…
marty cohen
- 110,916
124
votes
18 answers
How can I understand and prove the "sum and difference formulas" in trigonometry?
The " sum and difference " formulas often come in handy, but it's not immediately obvious that they would be true. \begin{align} \sin(\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\ \cos(\alpha \pm \beta) &= \cos \alpha…
Tyler
- 3,167
124
votes
1 answer
Continuous projections on $\ell_1$ with norm $>1$
I was trying to find papers and articles about non-contractive continuous projections on $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case $S=\mathbb{N}$. I've found one quite general…
Norbert
- 58,669
124
votes
11 answers
Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$
I've been looking at $$\int\limits_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$$ It seems that it always evaluates in terms of $\sin X$ and $\pi$, where $X$ is to be determined. For example: $$\displaystyle \int\limits_0^\infty {\frac{{{x^1}}}{{1 +…
Pedro
- 125,989
123
votes
14 answers
Can you give an example of a complex math problem that is easy to solve?
I am working on a project presentation and would like to illustrate that it is often difficult or impossible to estimate how long a task would take. I’d like to make the point by presenting three math problems (proofs, probably) that on the surface…
Judy
- 1,341
123
votes
2 answers
Making Friends around a Circular Table
I have $n$ people seated around a circular table, initially in arbitrary order. At each step, I choose two people and switch their seats. What is the minimum number of steps required such that every person has sat either to the right or to the left…
Vincent Tjeng
- 3,324
123
votes
8 answers
Lebesgue integral basics
I'm having trouble finding a good explanation of the Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Let's take for example a function $f(x) = x^2$. How do we…
user957
- 3,477
123
votes
9 answers
Proof of Frullani's theorem
How can I prove the Theorem of Frullani? I did not even know all the hypothesis that $f$ must satisfy, but I think that this are Let $\,f:\left[ {0,\infty } \right) \to \mathbb R$ be a a continuously differentiable function such that $$ \mathop…
August
- 3,643