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1500 questions
102
votes
4 answers
Alternative proof that $\frac{a^2+b^2}{ab+1}$ is a square when it's an integer.
Let $a,b$ be positive integers. When $$k = \frac{a^2 + b^2}{ab+1}$$ is an integer, it is a square. Proof 1 : (Ngô Bảo Châu): Rearrange to get $a^2-akb+b^2-k=0$, as a quadratic in $a$ this has two values: $a$ and $kb - a = (b^2-k)/a$. (The second…
quanta
- 12,783
102
votes
17 answers
How do you find the center of a circle with a pencil and a book?
Given a circle on a paper, and a pencil and a book. Can you find the center of the circle with the pencil and the book?
zdd
- 1,053
102
votes
4 answers
Are the eigenvalues of $AB$ equal to the eigenvalues of $BA$?
First of all, am I being crazy in thinking that if $\lambda$ is an eigenvalue of $AB$, where $A$ and $B$ are both $N \times N$ matrices (not necessarily invertible), then $\lambda$ is also an eigenvalue of $BA$? If it's not true, then under what…
dantswain
- 1,185
102
votes
39 answers
Unusual mathematical terms
From time to time, I come across some unusual mathematical terms. I know something about strange attractors . I also know what Witch of Agnesi is. However, what prompted me to write this question is that I was really perplexed when I read the other…
VividD
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102
votes
4 answers
A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational
Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational. What I have tried: Denote $x^n=r$ and $(x+1)^n=s$ with $r$, $s$ rationals. For each $k=0,1,\ldots, n−2$…
Dan Ismailescu
- 1,203
101
votes
9 answers
Prove that the union of countably many countable sets is countable.
I am doing some homework exercises and stumbled upon this question. I don't know where to start. Prove that the union of countably many countable sets is countable. Just reading it confuses me. Any hints or help is greatly appreciated! Cheers!
wonggr
- 1,783
101
votes
2 answers
Sheaf cohomology: what is it and where can I learn it?
As I understand it, sheaf cohomology is now an indispensable tool in algebraic geometry, but was originally developed to solve problems in algebraic topology . I have two questions about the matter. Question 1. What is sheaf cohomology? I have a…
Zhen Lin
- 98,824
101
votes
3 answers
What is "ultrafinitism" and why do people believe it?
I know there's something called "ultrafinitism" which is a very radical form of constructivism that I've heard said means people don't believe that really large integers actually exist. Could someone make this a little bit more precise? Are there…
Noah Snyder
- 10,303
101
votes
8 answers
What is $\gcd(0,0)$?
What is the greatest common divisor of $0$ and $0$? On the one hand, Wolfram Alpha says that it is $0$ ; on the other hand, it also claims that $100$ divides $0$ , so $100$ should be a greater common divisor of $0$ and $0$ than $0$.
Konstantin Weitz
- 1,295
101
votes
2 answers
Do most numbers have exactly $3$ prime factors?
In this question I plotted the number of numbers with $n$ prime factors. It appears that the further out on the number line you go, the number of numbers with $3$ prime factors get ahead more and more. The charts show the number of numbers with…
101
votes
11 answers
What's so special about standard deviation?
Equivalently, about variance? I realize it measures the spread of a distribution, but many other metrics could do the same (e.g., the average absolute deviation). What is its deeper significance? Does it have a particular geometric interpretation…
blue note
- 1,365
101
votes
10 answers
"Where" exactly are complex numbers used "in the real world"?
I've always enjoyed solving problems in the complex numbers during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be used/applied and hence am curious. So what…
PhD
- 2,881
101
votes
7 answers
101
votes
11 answers
Does notation ever become "easier"?
I'm in my first semester of college going for a math major and it's pretty great. I'm doing well, however, there seems to be huge gap between how difficult /complex an idea is and how convoluted it is presented . Let me make an example: In Analysis…
Jonathan
- 3,140
101
votes
15 answers
Example of a very simple math statement in old literature which is (verbatim) a pain to understand
As you know, before symbolic notations were introduced and adopted by the mathematical community, even simple statements were written in a very complicated manner because the writer had (nearly) only words to describe an equation or a result. What…
Cauchy
- 4,139