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1500 questions
111
votes
5 answers
Fibonacci number that ends with 2014 zeros?
This problem is giving me the hardest time:
Prove or disprove that there is a Fibonacci number that ends with 2014 zeros.
I tried mathematical induction (for stronger statement that claims that there is a Fibonacci number that ends in any number…
VividD
- 15,966
111
votes
5 answers
Difference between "≈", "≃", and "≅"
In mathematical notation, what are the usage differences between the various approximately-equal signs "≈", "≃", and "≅"?
The Unicode standard lists all of them inside the Mathematical Operators Block.
≈ : ALMOST EQUAL TO (U+2248)
≃ :…
GOTO 0
- 1,802
111
votes
7 answers
Variance of sample variance?
What is the variance of the sample variance? In other words I am looking for $\mathrm{Var}(S^2)$.
I have started by expanding out $\mathrm{Var}(S^2)$ into $E(S^4) - [E(S^2)]^2$
I know that $[E(S^2)]^2$ is $\sigma$ to the power of 4. And that is as…
MathMan
- 1,319
111
votes
6 answers
Finite subgroups of the multiplicative group of a field are cyclic
In Grove's book Algebra, Proposition 3.7 at page 94 is the following
If $G$ is a finite subgroup of the multiplicative group $F^*$ of a field $F$,
then $G$ is cyclic.
He starts the proof by saying "Since $G$ is the direct product of its Sylow…
QETU
- 1,119
111
votes
5 answers
What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into?
What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into?
The image below is a flawed example, from http://www.mathpuzzle.com/flawed456075.gif
Laczkovich gave a solution with many hundreds of…
Ed Pegg
- 20,955
111
votes
3 answers
Is the derivative the natural logarithm of the left-shift?
(Disclaimer: I'm a high school student, and my knowledge of mathematics extends only to some elementary high school calculus. I don't know if what I'm about to do is valid mathematics.)
I noticed something really neat the other day.
Suppose we…
David Zhang
- 8,835
111
votes
6 answers
What would base $1$ be?
Base $10$ uses these digits: $\{0,1,2,3,4,5,6,7,8,9\};\;$ base $2$ uses: $\{0,1\};\;$ but what would base $1$ be?
Let's say we define Base $1$ to use: $\{0\}$.
Because $10_2$ is equal to $010_2$, would all numbers be equal?
The way I have thought…
Justin
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111
votes
3 answers
Grothendieck 's question - any update?
I was reading Barry Mazur's biography and come across this part:
Grothendieck was exceptionally patient with me, for when we first met I knew next to nothing about algebra. In one of his first conversations with me, he raised the question (asked of…
Bombyx mori
- 19,638
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- 112
111
votes
9 answers
What are Some Tricks to Remember Fatou's Lemma?
For a sequence of non-negative measurable functions $f_n$, Fatou's lemma is a statement about the inequality
$$\int \liminf_{n\rightarrow \infty} f_n \mathrm{d}\mu \leq \liminf_{n\rightarrow \infty}(\int f_n \mathrm{d} \mu)$$
or alternatively (for…
Learner
- 7,278
111
votes
1 answer
Is this continuous analogue to the AM–GM inequality true?
First let us remind ourselves of the statement of the AM–GM inequality:
Theorem: (AM–GM Inequality) For any sequence $(x_n)$ of $N\geqslant 1$ non-negative real numbers, we have $$\frac1N\sum_k x_k \geqslant \left(\prod_k x_k\right)^{\frac1N}$$
It…
user1892304
- 2,808
111
votes
1 answer
Finding primes so that $x^p+y^p=z^p$ is unsolvable in the $p$-adic units
On my number theory exam yesterday, we had the following interesting problem related to Fermat's last theorem:
Suppose $p>2$ is a prime. Show that $x^p+y^p=z^p$ has a solution in $\mathbb{Z}_p^{\times}$ if and only if there exists an integer $a$…
ArtW
- 3,495
111
votes
1 answer
Tate conjecture for Fermat varieties
I've been looking at Tate's Algebraic Cycles and Poles of Zeta Functions (hard to find online... Google books outline here) and have a question about his work on (conjecturing!) the Tate conjecture for the Fermat variety $X_m^r$, defined by the…
Alex
- 4,108
111
votes
6 answers
Why don't analysts do category theory?
I'm a mathematics student in abstract algebra and algebraic geometry. Most of my books cover a great deal of category theory and it is an essential tool in understanding these two subjects.
Recently, I started taking some functional analysis courses…
gary
- 1,111
110
votes
16 answers
Good book for self study of functional analysis
I am a EE grad. student who has had one undergraduate course in real analysis (which was pretty much the only pure math course that I have ever done). I would like to do a self study of some basic functional analysis so that I can be better prepared…
EVK
- 1,259
110
votes
18 answers
Mathematical equivalent of Feynman's Lectures on Physics?
I'm slowly reading through Feynman's Lectures on Physics and I find myself wondering, is there an analogous book (or books) for math? By this, I mean a good approach to mathematics given through sweeping motions, appeals to intuition and an…
Cotton Seed
- 423