Most Popular
1500 questions
75
votes
7 answers
Is an automorphism of the field of real numbers the identity map?
Is an automorphism of the field of real numbers $\mathbb{R}$ the identity map?
If yes, how can we prove it?
Remark An automorphism of $\mathbb{R}$ may not be continuous.
Makoto Kato
- 42,602
75
votes
14 answers
Good books on mathematical logic?
I just started to learn mathematical logic. I'm a graduate student. I need a book with relatively more examples. Any recommendation?
ciciplus
75
votes
14 answers
"Naturally occurring" non-Hausdorff spaces?
It is not difficult for a beginning point-set topology student to cook up an example of a non-Hausdorff space; perhaps the simplest example is the line with two origins. It is impossible to separate the two origins with disjoint open sets.
It is…
Eric
- 1,632
75
votes
1 answer
About Euclid's Elements and modern video games
Update (6/19/2014) $\;$ Just wanted to say that this idea that I posted more than a year ago, has now become reality at: http://euclidthegame.com/
12.292 users have played it in 96 different countries, and 1232 people have reached level 20 :)
Update…
Kasper
- 13,528
75
votes
3 answers
How did Euler prove the Mersenne number $2^{31}-1$ is a prime so early in history?
I read that Euler proved $2^{31} -1$ is prime. What techniques did he use to prove this so early on in history? Isn't very large number stuff done with computers? Do you know if Euler had a team of people to follow algorithms for him, dubbed…
SwimBikeRun
- 1,047
- 1
- 10
- 18
75
votes
14 answers
What are some mathematical topics that involve adding and multiplying pictures?
Let me give you an example of what I mean. Flag algebras are a tool used in extremal graph theory which involve writing inequalities that look like:
(It's not too important to my question what this inequality means, but let me give you some…
Misha Lavrov
- 142,276
75
votes
4 answers
$\sum k! = 1! +2! +3! + \cdots + n!$ ,is there a generic formula for this?
I came across a question where I needed to find the sum of the factorials of the first $n$ numbers. So I was wondering if there is any generic formula for this?
Like there is a generic formula for the series:
$$ 1 + 2 + 3 + 4 + \cdots + n =…
vikiiii
- 2,681
- 2
- 34
- 45
75
votes
14 answers
What is $x^y$? How to understand it?
$x+y=z$
I have a pen. He has a pen. Total is two pen. This is plus.
$x-y=z$
I had two pens. A pen was lost. So, I have a pen. Total remaining is one. This is minus.
$x\cdot y=z$
I have two pens. Three other friends have two pens. $4…
Spacez Ly Wang
- 851
75
votes
1 answer
Divisor -- line bundle correspondence in algebraic geometry
I know a little bit of the theory of compact Riemann surfaces, wherein there is a very nice divisor -- line bundle correspondence.
But when I take up the book of Hartshorne, the notion of Cartier divisor there is very confusing. It is certainly not…
user977
75
votes
9 answers
Why does the symbol for the multiplication operation change shape?
Why does the "$\times$" used in arithmetic change to a "$\cdot$" as we progress through education? The symbol seems to only be ambiguous because of the variable $x$; however, we wouldn't have chosen the variable $x$ unless we were already removing…
user64742
- 2,207
75
votes
5 answers
How to write a good mathematical paper?
I hesitate to ask this question. However I read many advices from math.stackexchange, and I couldn't find anything similar.
A good time always goes too fast! Two years are fled. In the third year of PHD, my major is general topology and I'm facing…
Paul
- 20,553
75
votes
10 answers
Highest power of a prime $p$ dividing $N!$
How does one find the highest power of a prime $p$ that divides $N!$ and other related products?
Related question: How many zeros are there at the end of $N!$?
This is being done to reduce abstract duplicates. See
Coping with *abstract* duplicate…
user17762
75
votes
3 answers
how to read a mathematical paper?
I hope that this question is on-topic, though it is not quite technical.
I am curious to hear from people how they approach reading a mathematical paper.
I am not asking specific questions on purpose, though at first I had a few. But I want to keep…
normvector
- 713
74
votes
5 answers
A new imaginary number? $x^c = -x$
Being young, I don't have much experience with imaginary numbers outside of the basic usages of $i$. As I was sitting in my high school math class doing logs, I had an idea of something that would allow solving for logs with negative bases or with…
Warren L.
- 853
74
votes
2 answers
Combinatorial proof that $\sum \limits_{k=0}^n \binom{2k}{k} \binom{2n-2k}{n-k} (-1)^k = 2^n \binom{n}{n/2}$ when $n$ is even
In my answer here I prove, using generating functions, a statement equivalent to
$$\sum_{k=0}^n \binom{2k}{k} \binom{2n-2k}{n-k} (-1)^k = 2^n \binom{n}{n/2}$$
when $n$ is even. (Clearly the sum is $0$ when $n$ is odd.) The nice expression on the…
Mike Spivey
- 55,550