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1500 questions
184
votes
11 answers
How far can one get in analysis without leaving $\mathbb{Q}$?
Suppose you're trying to teach analysis to a stubborn algebraist who refuses to acknowledge the existence of any characteristic $0$ field other than $\mathbb{Q}$. How ugly are things going to get for him? The algebraist argues that the real numbers…
Alexander Gruber
- 28,325
184
votes
3 answers
Is $2048$ the highest power of $2$ with all even digits (base ten)?
I have a friend who turned $32$ recently. She has an obsessive compulsive disdain for odd numbers, so I pointed out that being $32$ was pretty good since not only is it even, it also has no odd factors. That made me realize that $64$ would be an…
Brian Rothstein
- 1,943
183
votes
12 answers
Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
For all $a, m, n \in \mathbb{Z}^+$, $$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$
Juan Liner
183
votes
7 answers
Induction on Real Numbers
One of my Fellows asked me whether total induction is applicable to real numbers, too ( or at least all real numbers ≥ 0) . We only used that for natural numbers so far. Of course you have to change some things in the inductive step, when you want…
Baju
- 1,933
183
votes
14 answers
How do you revise material that you already half-know, without getting bored and demotivated?
Mathematics inevitably involves a lot of self-teaching; if you're just planning to sit there and wait for the lecturer to introduce you to important ideas, you probably need to find yourself another career. So, like a lot people here, I try to…
goblin GONE
- 69,755
182
votes
2 answers
Open problems in General Relativity
I would like to know if there are some open mathematical problems in General Relativity, that are important from the point of view of Physics. Is there something that still needs to be justified mathematically in order to have solid foundations?
Benjamin
- 2,924
182
votes
14 answers
How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?
It is well known that $\sqrt{2}$ is irrational, and by modifying the proof (replacing even with divisible by $3$ ), one can prove that $\sqrt{3}$ is irrational, as well. On the other hand, clearly $\sqrt{n^2} = n$ for any positive integer $n$ . It…
anonymous
182
votes
22 answers
Striking applications of integration by parts
What are your favorite applications of integration by parts? (The answers can be as lowbrow or highbrow as you wish. I'd just like to get a bunch of these in one place!) Thanks for your contributions, in advance!
Jon Bannon
- 3,233
180
votes
17 answers
Alternative notation for exponents, logs and roots?
If we have $$ x^y = z $$ then we know that $$ \sqrt[y]{z} = x $$ and $$ \log_x{z} = y .$$ As a visually-oriented person I have often been dismayed that the symbols for these three operators look nothing like one another, even though they all tell us…
friedo
- 2,863
180
votes
26 answers
Software for drawing geometry diagrams
What software do you use to accurately draw geometry diagrams ?
Lucky
- 1,159
180
votes
9 answers
Why is $1^{\infty}$ considered to be an indeterminate form
From Wikipedia : In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by…
user9413
179
votes
10 answers
Lesser-known integration tricks
I am currently studying for the GRE math subject test, which heavily tests calculus. I've reviewed most of the basic calculus techniques (integration by parts, trig substitutions, etc.) I am now looking for a list or reference for some lesser-known…
GREStudying12345
- 271
179
votes
13 answers
What is the best book to learn probability?
Question is quite straight... I'm not very good in this subject but need to understand at a good level.
Eduardo Xavier
- 148
178
votes
2 answers
Proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ (and now $\pi^{\left(\pi^{\pi^\pi}\right)}$) is a noninteger.
Conor McBride asks for a fast proof that $$x = {\left(\pi^\pi\right)}^{\pi^\pi}$$ is not an integer. It would be sufficient to calculate a very rough approximation, to a precision of less than $1,$ and show that $n < x < n+1$ for some integer $n$ .…
MJD
- 68,126
- 44
- 311
- 628
178
votes
7 answers
Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$
I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$. I found a solution by myself 10 hours after I posted it, here it is: $$f(x)=\sum_{\substack{n=1\\n\text{…
danodare
- 1,925