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1500 questions
171
votes
1 answer
What's the significance of Tate's thesis?
I've just sat through several lectures that proved most of the results in Tate's thesis: the self-duality of the adeles, the construction of "zeta functions" by integration, and the proof of the functional equation. However, while I was able to…
Akhil Mathew
- 32,250
170
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
- 1,591
170
votes
2 answers
Example of infinite field of characteristic $p\neq 0$
Can you give me an example of infinite field of characteristic $p\neq0$?
Thanks.
Aspirin
- 5,889
170
votes
7 answers
Find a real function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$?
I've been perusing the internet looking for interesting problems to solve. I found the following problem and have been going at it for the past 30 minutes with no success:
Find a function $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(f(x)) = -x$ for…
Gamma Function
- 5,752
170
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
- 147,345
170
votes
19 answers
What actually is a polynomial?
I can perform operations on polynomials. I can add, multiply, and find their roots. Despite this, I cannot define a polynomial.
I wasn't in the advanced mathematics class in 8th grade, then in 9th grade I skipped the class and joined the more…
Travis
- 3,604
169
votes
7 answers
Is non-standard analysis worth learning?
As a former physics major, I did a lot of (seemingly sloppy) calculus using the notion of infinitesimals. Recently I heard that there is a branch of math called non-standard analysis that provides some formalism to this type of calculus.
So, do you…
user13255
169
votes
7 answers
Is infinity an odd or even number?
My 6 year old wants to know if infinity is an odd or even number. His 38 year old father is keen to know too.
Kevin
- 1,701
169
votes
17 answers
Why does factoring eliminate a hole in the limit?
$$\lim _{x\rightarrow 5}\frac{x^2-25}{x-5} = \lim_{x\rightarrow 5} (x+5)$$
I understand that to evaluate a limit that has a zero ("hole") in the denominator we have to factor and cancel terms, and that the original limit is equal to the new and…
Emi Matro
- 5,103
169
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
- 468,795
168
votes
15 answers
Monty hall problem extended.
I just learned about the Monty Hall problem and found it quite amazing. So I thought about extending the problem a bit to understand more about it.
In this modification of the Monty Hall Problem, instead of three doors, we have four (or maybe $n$)…
Shaurya Gupta
- 4,449
168
votes
1 answer
Classification of prime ideals of $\mathbb{Z}[X]$
Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable over $\Bbb Z$.
My question: Is every prime ideal of $\mathbb{Z}[X]$ one of following types?
If yes, how would you prove this?
$(0)$.
$(f(X))$, where $f(X)$ is an irreducible…
Makoto Kato
- 44,216
167
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
- 1,799
- 2
- 13
- 24
167
votes
9 answers
Intuitive interpretation of the Laplacian Operator
Just as the gradient is "the direction of steepest ascent", and the divergence is "amount of stuff created at a point", is there a nice interpretation of the Laplacian Operator (a.k.a. divergence of gradient)?
koletenbert
- 4,150
167
votes
1 answer
Overview of basic facts about Cauchy functional equation
The Cauchy functional equation asks about functions $f \colon \mathbb R \to \mathbb R$ such that
$$f(x+y)=f(x)+f(y).$$
It is a very well-known functional equation, which appears in various areas of mathematics ranging from exercises in freshman…
Martin Sleziak
- 56,060