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1500 questions
108
votes
8 answers
Show that the set of all finite subsets of $\mathbb{N}$ is countable.
Show that the set of all finite subsets of $\mathbb{N}$ is countable. I'm not sure how to do this problem. I keep trying to think of an explicit formula for 1-1 correspondence like adding all the elements in each subset and sending that sum to…
user39794
108
votes
2 answers
GRE Subject Test - Past Papers, Books, Advice
This is not for the Maths part of the General GRE. This is for the GRE Subject Test in Maths . Feel free to add or comment. How do I know the definition of rings or of anything on the GRE given that definitions can vary? What does the subject GRE…
BCLC
- 14,383
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
3 answers
Cardioid in coffee mug?
I've been learning about polar curves in my Calc class and the other day I saw this suspiciously $r=1-\cos \theta$ looking thing in my coffee cup (well actually $r=1-\sin \theta$ if we're being pedantic.) Some research revealed that it's called a…
meiji163
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107
votes
9 answers
Congruence Arithmetic Laws, e.g. in divisibility by $7$ test
I have seen other criteria for divisibility by $7$ . Criterion described below present in the book Handbook of Mathematics for IN Bronshtein (p. $323$ ) is interesting, but could not prove it. Let $n = (a_ka_{k-1}\ldots a_2a_1a_0)_{10} =…
Mathsource
- 5,663
107
votes
21 answers
For any prime $p > 3$, why is $p^2-1$ always divisible by $24$?
Let $p>3$ be a prime. Prove that $24 \mid p^2-1$ . I know this is very basic and old hat to many, but I love this question and I am interested in seeing whether there are any proofs beyond the two I already know.
Don De Tina
- 1,239
107
votes
27 answers
Mathematicians ahead of their time?
It is said that in every field, there is that person who was years ahead of their time. For instance, Paul Morphy is said to have retired from chess because he found no one to match his technique, which very much resembled modern chess theory. So,…
hb20007
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107
votes
1 answer
Geometric interpretation of the Riemann-Roch for curves
Let $X$ be a smooth projective curve of genus $g\geq2$ over an algebraically closed field $k$ and denote by $K$ a canonical divisor. I have some clues about the geometrical interpretation of the Riemann-Roch Theorem for smooth algebraic curves, but…
Abramo
- 7,173
107
votes
8 answers
What is a universal property?
Sorry, but I do not understand the formal definition of "universal property" as given at Wikipedia . To make the following summary more readable I do equate "universal" with "initial" and omit the tedious details concerning duality. Suppose that $U:…
Hans-Peter Stricker
- 18,829
107
votes
1 answer
All polynomials with no natural roots and integer coefficients such that $\phi(n)|\phi(P(n))$
Let $P(x)$ be a polynomial with integer coefficients such that the equation $P(x)=0$ has no positive integer solutions. Find all polynomials $P(x)$ such that for all positive integers $n$ we have $\phi(n) \mid \phi(P(n))$. It is conjectured there…
Amir Parvardi
- 4,946
107
votes
7 answers
Is the "determinant" that shows up accidental?
Consider the class of rational functions that are the result of dividing one linear function by another: $$\frac{a + bx}{c + dx}$$ One can easily compute that, for $\displaystyle x \neq \frac cd$ $$\frac{\mathrm d}{\mathrm dx}\left(\frac{a + bx}{c +…
stats_model
- 1,171
107
votes
11 answers
Riddles that can be solved by meta-assumptions
The website of my university posted a riddle that goes something like this: Riddle There are three men named 1,2 and 3 and each one has two colored dots on his forehead. Possible colors are black and red . No color is used more than four times . The…
M. Winter
- 30,997
107
votes
4 answers
Importance of matrix rank
What is the importance of the rank of a matrix? I know that the rank of a matrix is the number of linearly independent rows or columns (whichever is smaller). Why is it a problem if a matrix is rank deficient? Also, why is the smaller value between…
0x0
- 2,711
107
votes
8 answers
Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$
Compute $$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$$
user 1591719
- 45,325
107
votes
2 answers
Is there really no way to integrate $e^{-x^2}$?
Today, in my calculus class, we encountered the function $e^{-x^2}$ , and I was told that it was not integrable. I was very surprised. Is there really no way to find the integral of $e^{-x^2}$ ? Graphing $e^{-x^2}$ , it appears as though it should…
Zolani13
- 1,981