22
votes
Accepted
Simplest diophantine equation with open solvability
Determining which integers $n$ are a sum of three cubes is a very famous open problem:
$$a^3 + b^3 + c^3 = n, \quad a,b,c \in \mathbb{Z}.$$
Conjecturally, $n$ is a sum of three cubes iff $n \not \...
17
votes
Is there any published summary of Erdos's published problems in the American Mathematical Monthly journal?
I do not know of such a summary. But JSTOR has spent considerable time indexing the Monthly, including the "Problems and Solutions" sections. So if your library has a JSTOR subscription you can ...
15
votes
Characterization of a sphere: every "sub-sphere" has two centers
Blaschke's conjecture might be relevant. In particular, if the surface is $S^2$ with a $C^3$ metric, then it follows from Blaschke's conjecture, proved by Green. This states that a metric on the ...
14
votes
Accepted
Covering the disk with a family of infinite total measure
No even in dimension 1 (and multiplying the example for $\mathbb{R}$ by the small segment you get a counterexample in $\mathbb{R}^2$).
Take the set $A_n\subset \mathbb{R}$ defined as $\bigcup_{k\in \...
13
votes
Simplest diophantine equation with open solvability
It's more complicated than the other answers by MattF and DanielLoughran, but the Erdős–Straus conjecture states that for every integer $n \ge 2$, there exist positive integers $x, y, z$ such that
$$\...
9
votes
Accepted
Inequality with symmetric polynomials
This looks like a better fit for Math Stackexchange, because it's
the kind of thing one learns from Olympiad problem books . . .
One standard approach that has not been mentioned yet:
We may assume $...
6
votes
Contest problems with connections to deeper mathematics
$\textbf{Problem IMO 1986}$. Let $n$ be a natural number. If $k^{2}+k+n$ is a prime number for $0\leq k \leq \lfloor\sqrt{n/3}\rfloor$, then show that $k^{2}+k+n$ is a prime for $0\leq k \leq n-2$.
...
Community wiki
5
votes
Accepted
Where can I access American Mathematical Monthly problems given an index?
Unfortunately, I don't think there's any particularly easy way to find a specific problem given its index number, but let me summarize some of the comments (and add some of my own) in a community wiki ...
Community wiki
5
votes
Examples of using physical intuition to solve math problems
My favorite example: it's not too hard to show---see here for example---that if $U$ is a unitary matrix, then $\left|\operatorname{Per}(U)\right|\le1$, where Per denotes the matrix permanent function,
...
Community wiki
4
votes
Inequality with symmetric polynomials
This is easily proven using the Rearrangement Inequality, which says that if we have two sequences of reals, and we are to pair them up in such a way as to maximize the sum of the products of the ...
4
votes
Inequality with symmetric polynomials
Follows from Hölder's inequality (p=6, q = 6/5):
$ab^5 + ba^5 \le (a^6+b^6)^{1/6} (b^6+a^6)^{5/6}$
4
votes
Is there any published summary of Erdos's published problems in the American Mathematical Monthly journal?
I do not think that anyone ever made an effort to collect ALL problems of Erdos.
In the later part of his life he liked to give talks on various conferences under the title "My favorite unsolved ...
4
votes
Accepted
Covering the disk with a family of infinite total measure - the convex sequel
The planar case is rather simple but I'm still struggling with dimension 3 and higher, so we'd better keep this thread open at least for a while.
Lemma 1: Suppose we have finitely many infinite ...
4
votes
Method to solve system of exponential sums of the form $a^x+b^x=c$ given more equations than variables
I'm not sure about your method, but such equations can easily be turned into polynomial equations by introducing $u:=a^{x-1}$ and $v:=b^{x-1}$:
\begin{cases}
u+v=337 \\
au+bv=1267 \\
a^2u+b^2v=4825 \\
...
3
votes
Accepted
Classifying functions up to suitable pre-composition and/or post-composition
The emphasis on finding solutions to equations $f(x) = y$ is a red herring, and note that in your examples your symmetries aren't acting pointwise on $Y$ as in your general explanation (sometimes they ...
3
votes
Inequality with symmetric polynomials
This is a very very special case of Muirhead's inequality. It involves two multi-exponents $\newcommand{\bZ}{\mathbb{Z}}$ $\alpha,\beta\in\bZ_{\geq 0}^n$ and the natural action of the symmetric ...
3
votes
Inequality with symmetric polynomials
Also follows from Muirhead's inequality since $(6,0)$ majorizes $(5,1)$.
3
votes
Accepted
the sum of fractional parts times the ordinary powers
As Gerry already commented, your formulas are equivalent to a reciprocity theorem of Tom Apostol; see, e.g., the bibliography in this paper (Apostol's paper is too old to be on the arXiv): Your sums ...
2
votes
List of recently solved mathematical problems
Here https://theorems.home.blog/theorems-list/ is the website you are asking for.
It covers all recently solved mathematical problems, which are important (for example, published in a top journal) ...
Community wiki
2
votes
Accepted
Characterize the Monotonicity of a root of a cubic equation
Differentiating $h(x, \eta) = 0$ implicitly, we get
$$ \dfrac{dx}{d\eta} = \dfrac{-c\eta^2 + V(2\eta+x)(\eta-x)^2}{\eta (3 V (\mu - x)^2 + c \eta)}$$
The denominator is always positive.
On any ...
2
votes
A problem with elementary inequality involving probabilities and Brier scoring rule
Clearly $\sum_{i=1}^{n-1}p_i - (\sum_{i=1}^{n-1}p_i)^2\geq 0$, so we can ignore those terms in the inequality. Since $(\frac{1}{1-p_n})\sum_{i=1}^{n-1}p_i^2 \geq \sum_{i=1}^{n-1}p_i^2$ it suffices to ...
2
votes
Contest problems with connections to deeper mathematics
IMO 2003-6:
Show that for each prime $p$, there exists a prime $q$ such that $n^p − p$ is not divisible by $q$ for any positive integer $n$.
This obviously looks like something you want to apply the ...
2
votes
Accepted
Method to solve system of exponential sums of the form $a^x+b^x=c$ given more equations than variables
We can use linear algebra to make this easier.
Consider the space of sequences of the form $A a^n + B b^n$. We can use several dual bases for this space. One is the obvious basis of maps that send $A ...
2
votes
Accepted
Solving an recursive sequence
$$f(x+1) = f(x)^2 - 2,\;\;f(0)=4$$
$$\Rightarrow f(x)=2\cos\left(2^x\arccos 2\right)=2 \cosh \left(2^x \ln \left(\sqrt{3}+2\right)\right)$$
1
vote
Is it possible to find UNSATisfiable solutions to a SAT problem with a SAT problem?
Indeed, it is most likely not possible to find a polynomial-size propositional formula whose models are exactly the non-outputs of $f(x,y)$, the problem for general $f$ is $\Sigma_2^p$-complete. You ...
1
vote
Contest problems with connections to deeper mathematics
Colorado Mathematical Olympiad (1986): Santa Claus and his elves paint the plane with two colors, red and green. Prove that the plane contains two points of the same color, exactly one inch apart.
So ...
Community wiki
1
vote
Characterize the Monotonicity of a root of a cubic equation
In reply to the follow-up questions from FTXX, here is an argument. Using Israel's notations, suppose $h(x,\eta)=0$ and $\frac{dx}{d\eta}=0$. That means
$$V(\eta-x)^3=c\eta(\eta+x) \tag1$$
and
$$-c\...
1
vote
Examples of using physical intuition to solve math problems
Let A, B, C be three points in the plane and assume that P minimizes the sum of the distances to A, B, C. One can proof that then the angles APB, BPC, CPA are all 120 deg. Physically we can imagine a ...
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