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1500 questions
172
votes
1 answer
Does every ring of integers sit inside a ring of integers that has a power basis?
Given a finite extension of the rationals, $K$, we know that $K=\mathbb{Q}[\alpha]$ by the primitive element theorem, so every $x \in K$ has the form
$$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n,$$
with $a_i \in \mathbb{Q}$.
However, the ring of…
Eins Null
- 2,127
172
votes
1 answer
Is there a categorical definition of submetry?
(Updated to include effective epimorphism.)
This question is prompted by the recent discussion of why analysts don't use category theory. It demonstrates what happens when an analyst tries to use category theory.
Consider the category CpltMet in…
user31373
171
votes
4 answers
The Hole in One Pizza
In a recent issue of Crux, at the end of the editorial (which is public), it appears the following very nice problem by Peter Liljedahl.
I couldn't resist sharing it with the MSE community. Enjoy!
Robert Z
- 145,942
171
votes
45 answers
Is there any integral for the Golden Ratio?
I was wondering about important/famous mathematical constants, like $e$, $\pi$, $\gamma$, and obviously the golden ratio $\phi$.
The first three ones are really well known, and there are lots of integrals and series whose results are simply those…
Enrico M.
- 26,114
170
votes
1 answer
Rational roots of polynomials
Can one construct a sequence $(a_k)_{k\geqslant 0}$ of rational numbers such that, for every positive integer $n$ the polynomial $a_nX^n+a_{n-1}X^{n-1}+\cdots +a_0$ has exactly $n$ distinct rational roots ?
If we cannot construct it explicitly, can…
user84673
- 2,027
170
votes
6 answers
Why is the Penrose triangle "impossible"?
I remember seeing this shape as a kid in school and at that time it was pretty obvious to me that it was "impossible". Now I looked at it again and I can't see why it is impossible anymore.. Why can't an object like the one represented in the…
user736690
170
votes
1 answer
Is there a homology theory that counts connected components of a space?
It is well-known that the generators of the zeroth singular homology group $H_0(X)$ of a space $X$ correspond to the path components of $X$.
I have recently learned that for Čech homology the corresponding statement would be that $\check{H}_0(X)$ is…
Dejan Govc
- 17,007
170
votes
34 answers
Can you provide me historical examples of pure mathematics becoming "useful"?
I am trying to think/know about something, but I don't know if my base premise is plausible. Here we go.
Sometimes when I'm talking with people about pure mathematics, they usually dismiss it because it has no practical utility, but I guess that…
Red Banana
- 23,956
- 20
- 91
- 192
170
votes
7 answers
What is the difference between Fourier series and Fourier transformation?
What's the difference between Fourier transformations and Fourier Series?
Are they the same, where a transformation is just used when its applied (i.e. not used in pure mathematics)?
Dean
- 1,881
168
votes
11 answers
Do we know if there exist true mathematical statements that can not be proven?
Given the set of standard axioms (I'm not asking for proof of those), do we know for sure that a proof exists for all unproven theorems? For example, I believe the Goldbach Conjecture is not proven even though we "consider" it true.
Phrased another…
Jeremy
- 1,571
168
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…
friedo
- 2,713
167
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
167
votes
1 answer
What properties of busy beaver numbers are computable?
The busy beaver function $\text{BB}(n)$ describes the maximum number of steps that an $n$-state Turing machine can execute before it halts (assuming it halts at all). It is not a computable function because computing it allows you to solve the…
Qiaochu Yuan
- 419,620
166
votes
1 answer
A variation of Fermat's little theorem in the form $a^{n-d}\equiv a$ (mod $p$).
Fermat's little theorem states that for $n$ prime,
$$
a^n \equiv a \pmod{n}.
$$
The values of $n$ for which this holds are the primes and the Carmichael numbers. If we modify the congruence slightly,
$$
a^{n - 1} \equiv a \pmod{n},
$$
the values of…
Tavian Barnes
- 1,789
166
votes
4 answers
What happens when we (incorrectly) make improper fractions proper again?
Many folks avoid the "mixed number" notation such as $4\frac{2}{3}$ due to its ambiguity. The example could mean "$4$ and two thirds", i.e. $4+\frac{2}{3}$, but one may also be tempted to multiply, resulting in $\frac{8}{3}$.
My questions pertain to…
Zim
- 4,318