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1500 questions
108
votes
23 answers
Open mathematical questions for which we really, really have no idea what the answer is
There is no shortage of open problems in mathematics. While a formal proof for any of them remains elusive, with the "yes/no" questions among them mathematicians are typically not working in both directions but rather have a pretty clear idea of…
user139000
108
votes
3 answers
Does this property characterize a space as Hausdorff?
As a result of this question, I've been thinking about the following condition on a topological space $Y$:
For every topological space $X$, $E\subseteq X$, and continuous maps $f,g\colon X\to Y$, if $E$ is dense in $X$, and $f$ and $g$ agree on $E$…
Arturo Magidin
- 398,050
108
votes
8 answers
Understanding of the theorem that all norms are equivalent in finite dimensional vector spaces
The following is a well-known result in functional analysis:
If the vector space $X$ is finite dimensional, all norms are equivalent.
Here is the standard proof in one textbook. First, pick a norm for $X$, say…
user9464
108
votes
0 answers
A question about divisibility of sum of two consecutive primes
I was curious about the sum of two consecutive primes and after proving that the sum for the odd primes always has at least 3 prime divisors, I came up with this question:
Find the least natural number $k$ so that there will be only a finite
number…
CODE
- 4,921
108
votes
6 answers
Does a non-trivial solution exist for $f'(x)=f(f(x))$?
Does $f'(x)=f(f(x))$ have any solutions other than $f(x)=0$?
I have become convinced that it does (see below), but I don't know of any way to prove this.
Is there a nice method for solving this kind of equation? If this equation doesn't have any…
hasnohat
- 3,882
- 1
- 18
- 26
108
votes
0 answers
Classification of local Artin (commutative) rings which are finite over an algebraically closed field
A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(\mathcal{A})\rightarrow X$ where $\mathcal A$ is a local Artin ring finite over $k$ can be extended to every $Y'\supset Y$ where $Y'$ is an infinitesimal thickening…
user16544
108
votes
6 answers
Self-studying real analysis — Tao or Rudin?
The reference requests for analysis books have become so numerous as to blot out any usefulness they could conceivably have had. So here comes another one.
Recently I've began to learn real analysis via Rudin. I would do all the exercises, and if I…
Lee Wang
- 1,859
108
votes
1 answer
Can we remove any prime number with this strange process?
This is a prime-removal algorithm I made, which may appear to be quite complex so I will start with an example. @Max has since added this sequence to OEIS, number A332198.
The process goes as follows:
Start with the first prime number,…
ə̷̶̸͇̘̜́̍͗̂̄︣͟
- 27,005
108
votes
8 answers
Are calculus and real analysis the same thing?
I guess this may seem stupid, but
how calculus and real analysis are
different from and related to each
other?
I tend to think they are the same
because all I know is that the
objects of both are real-valued
functions defined on $\mathbb{R}^n$,
and…
Tim
- 47,382
108
votes
11 answers
Riddles that can be solved by meta-assumptions
The website of my university posted a riddle that goes something like this:
Riddle
There are three men named 1,2 and 3 and each one has two colored dots on his forehead. Possible colors are black and red. No color is used more than four times. The…
M. Winter
- 29,928
108
votes
7 answers
What is the algebraic intuition behind Vieta jumping in IMO1988 Problem 6?
Problem 6 of the 1988 International Mathematical Olympiad notoriously asked:
Let $a$ and $b$ be positive integers and $k=\frac{a^2+b^2}{1+ab}$. Show that if $k$ is an integer then $k$ is a perfect square.
The usual way to show this involves a…
kdog
- 1,247
108
votes
5 answers
Difference between gradient and Jacobian
Could anyone explain in simple words (and maybe with an example) what the difference between the gradient and the Jacobian is?
The gradient is a vector with the partial derivatives, right?
Math_reald
- 1,317
107
votes
19 answers
Chatting about mathematics (with real-time LaTeX rendering)
Do you know about some tools which can be used for online chat about mathematics? In particular, I am interested in software which would be able to render LaTeX formulas. (Since LaTeX is probably the fastest possibility to type mathematics.)
Have…
Martin Sleziak
- 53,687
107
votes
3 answers
How prove this nice limit $\lim\limits_{n\to\infty}\frac{a_{n}}{n}=\frac{12}{\log{432}}$
Nice problem:
Let $a_{0}=1$ and
$$a_{n}=a_{\left\lfloor n/2\right\rfloor}+a_{\left\lfloor n/3 \right\rfloor}+a_{\left\lfloor n/6\right\rfloor}.$$ Show that
$$\lim_{n\to\infty}\dfrac{a_{n}}{n}=\dfrac{12}{\log{432}},$$
where $\lfloor x \rfloor$ is…
math110
- 93,304
107
votes
4 answers
Why is $e^{\pi \sqrt{163}}$ almost an integer?
The fact that Ramanujan's Constant $e^{\pi \sqrt{163}}$ is almost an integer ($262 537 412 640 768 743.99999999999925...$) doesn't seem to be a coincidence, but has to do with the $163$ appearing in it. Can you explain why it's almost-but-not-quite…
stevenvh
- 2,686