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1500 questions
123
votes
18 answers
Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$?
I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute $\sum_{n=1}^{\infty} \frac{1}{n^2}$?"
Are there any…
Mike Spivey
- 55,550
123
votes
3 answers
More than 99% of groups of order less than 2000 are of order 1024?
In Algebra: Chapter 0, the author made a remark (footnote on page 82), saying that more than 99% of groups of order less than 2000 are of order 1024.
Is this for real? How can one deduce this result? Is there a nice way or do we just check all…
Hui Yu
- 15,029
122
votes
2 answers
Making Friends around a Circular Table
I have $n$ people seated around a circular table, initially in arbitrary order. At each step, I choose two people and switch their seats. What is the minimum number of steps required such that every person has sat either to the right or to the left…
Vincent Tjeng
- 3,304
- 2
- 21
- 34
122
votes
13 answers
Good abstract algebra books for self study
Last semester I picked up an algebra course at my university, which unfortunately was scheduled during my exams of my major (I'm a computer science major). So I had to self study the material, however, the self written syllabus was not self study…
sxd
- 3,504
122
votes
27 answers
Different ways to prove there are infinitely many primes?
This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more…
Patrick Da Silva
- 41,413
122
votes
2 answers
Why is it hard to prove whether $\pi+e$ is an irrational number?
From this list I came to know that it is hard to conclude $\pi+e$ is an irrational? Can somebody discuss with reference "Why this is hard ?"
Is it still an open problem ? If yes it will be helpful to any student what kind ideas already used but…
users31526
- 1,999
- 3
- 13
- 13
122
votes
8 answers
What is the difference between a class and a set?
I know what a set is. I have no idea what a class is.
As best as I can make out, every set is also a class, but a class can be "larger" than any set (a so-called "proper class").
This obviously makes no sense whatsoever, since sets are of unlimited…
MathematicalOrchid
- 6,185
122
votes
19 answers
What is the most unusual proof you know that $\sqrt{2}$ is irrational?
What is the most unusual proof you know that $\sqrt{2}$ is irrational?
Here is my favorite:
Theorem: $\sqrt{2}$ is irrational.
Proof:
$3^2-2\cdot 2^2 = 1$.
(That's it)
That is a corollary of
this result:
Theorem:
If $n$ is a positive…
marty cohen
- 107,799
122
votes
1 answer
Continuous projections on $\ell_1$ with norm $>1$
I was trying to find papers and articles about non-contractive continuous projections on $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case $S=\mathbb{N}$.
I've found one quite general…
Norbert
- 56,803
121
votes
2 answers
What makes a theorem "fundamental"?
I've studied three so-called "fundamental" theorems so far (FT of Algebra, Arithmetic and Calculus) and I'm still unsure about what precisely makes them fundamental (or moreso than other theorems).
Wikipedia claims:
The fundamental theorem of a…
beep-boop
- 11,595
121
votes
15 answers
Infiniteness of non-twin primes.
Well, we all know the twin prime conjecture.
There are infinitely many primes $p$, such that $p+2$ is also prime.
Well, I actually got asked in a discrete mathematics course, to prove that there are infinitely many primes $p$ such that $p + 2$ is…
Tomas Wolf
- 1,351
121
votes
16 answers
Is 10 closer to infinity than 1?
This may be considered a philosophical question but is the number "10" closer to infinity than the number "1"?
termsofservice
- 1,049
121
votes
26 answers
Simplest or nicest proof that $1+x \le e^x$
The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or canonical proof? I would ideally like a proof which…
Ashley Montanaro
- 1,327
121
votes
7 answers
Strategies for Effective Self-Study
I have a long-term goal of acquiring graduate-level knowledge in Analysis, Algebra and Geometry/Topology. Once that is achieved, I am interested in applying this knowledge to both pure and applied mathematics. In particular, I am interested in…
ItsNotObvious
- 10,883
121
votes
13 answers
Do factorials really grow faster than exponential functions?
Having trouble understanding this. Is there anyway to prove it?
Billy Thompson
- 1,860
- 7
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- 23