While studying parametric identities involving sums of three cubes, I obtained the polynomial identity in 11 variables. The complete formulas and verification code are available here: https://zenodo.org/records/22255821 All these identities have been checked symbolically in SymPy. The parametric...

Let $X$ be an integral curve, $Y$ a smooth curve, and $f : X \to Y$ a finite surjective morphism. Given a vector bundle $E$ on $X$, what is $\operatorname{rk}(f_{*}E)$? Is it true that $$ \operatorname{rk}(f_{*}E)=\operatorname{rk}(E)\deg(f)? $$ If so, could you give me a proof or indicate a...

Let $A\subset\mathbb{N}$ have positive upper density. Must there exist infinitely many $n$ such that both $n$ and $n^2$ belong to $A$? If not, determine the largest possible upper density of a set $A$ for which $$ A\cap\{n^2:n\in A\} $$ is finite. Can anyone please help me with this?

Throughout, $\mathfrak S_n$ is assumed to be associated with a rank-$(n-2)$ zeta function. Consider $\mathfrak{S}_3 = (\mathcal{I}_3, \Gamma_3)$, composed of four orbifold sheets glued along a seam graph $$ \mathcal{I}_3 = \left(\bigsqcup_{i=1}^4 X_i\right)\Big/\!\sim_{\Gamma_3}, \qquad X_i =...

An interesting MSE post was recently made by Koushik Pramanik. To give some background, there seems to be only one known polynomial solution to the equation in the first part of the title, namely, $$(17 p^2 - 12 p q - 13 q^2)^4 + (17 p^2 + 12 p q - 13 q^2)^4 = (17 p^2 - q^2)^4 + (289 p^4 + 14...

Can someone relate the constants in the density function, of numbers whose digit sum divides its digit product to properties of these numbers? In attempting to solve Bernardo Recaman Santos' question on the Puzzling Stack Exchange, about triplets and quadruplets (in a row) of these numbers, as...

For the next multigrade diophantine chain $$x_1^k+x_2^k+x_3^k=y_1^k+y_2^k+y_3^k=z_1^k+z_2^k+z_3^k=N, k<5$$ This question is partly inspired by this similar question, but here we will look at degrees that are less than 5, which makes the discovery of new multi-grade solutions more likely....

$\DeclareMathOperator{\mod}{Mod} \def\O{\mathcal{O}} \def\F{\mathcal{F}} \def\G{\mathcal{G}} \DeclareMathOperator{\qcoh}{QCoh}$Let $X$ be a ringed space. We say that an $\O_X$-module $\F$ is (faithfully) flat if the endofunctor $(-)\otimes_{\O_X}\F$ on $\mod(X)$ is (faithfully) exact. It is...

(Note: I am aware about this post, my question is different) Proposition 4.20 in Görtz-Wedhorn states the following: All of the assumptions and assertions are local in $S,X,Y$ so we may assume they are affine (as we do in the proof of this proposition). However, except for injectivity of $g$,...

I have seen in some titles of articles phrase 'motivic category(ies)' but I haven't found definition. It seems to me that it's just some category of motives or more precisely subcategory of (un)stable motivic homotopy category.

I have been investigating finite-dimensional approaches to positivity criteria equivalent to the Riemann hypothesis, in particular Weil's criterion. Rather than assuming that such a criterion admits a finite reduction, I am trying to understand precisely what would be required to justify one....

Let $(a,b,c,d)$ be positive. In a previous question, we asked for parameterizations to, $$a^3+b^3+c^3 = (c+1)^3=d^3$$ where $c$ is a polynomial of deg-$n$ for $n>3$. Question: We've found $c$ of various deg-$n$, but now we give it sharper focus: Is there of deg-4 or deg-8? I. Degree 3 There are...

I have been using JSTOR to identify the original year, volume, issue, and problem number of problems that were cited only as “AMM” (American Mathematical Monthly) in various books. My approach has been to search for distinctive phrases and keywords from the problem statements and try to...

How can we show that the ideal $\mathfrak{p}:=(x^2-y,x^3-z) \subset k[x,y,z]$ is prime using the definition of prime ideals in commutative rings? To show the ideal is prime one defines a map $k[x,y,z] \to k[t]$ given by $f(x,y,x)\mapsto f(t,t^2,t^3)$, and shows that $k[x,y,z]/\mathfrak{p} \cong...

I'm interested in a particular relaxation of the magic square of squares problem, where the entries are of the form $a^2\pm\varepsilon$ for some $\varepsilon$. The original problem is when $\varepsilon=0$. By "correcting" the Parker square, there is a solution when $\varepsilon = 528$:...

Conjecture: Does the Diophantine equation $$x \cdot a^y + y \cdot b^z = z \cdot c^w + w \cdot d^x$$ And satisfying $$a + b + c + d = x + y + z + w$$ Also $$a \neq b \neq c \neq d \neq x \neq y \neq z \neq w$$ have any solutions in primes? Motivation: In a previous iteration of this problem,...

Hartshorne Proposition 1.5 states that any closed subset of a Noetherian topological space can be written as a union of finitely many closed irreducible subsets. A Noetherian topological space is a topological space that satisfies the descending chain condition on its closed subsets. I have one...

Motivation: While recently researching equations similar to this, I suddenly had an idea: would adding coefficients and specific restrictions change the difficulty? Therefore, I unexpectedly pieced together the following Diophantine equation: Let $a, b, c, d, x, y, z, w$ be 8 pairwise distinct...

The question is in the title: if you pick a natural number $j$, are you always guaranteed to have the fractional part of some $j^{k+1} \pi^k/2$ be strictly greater than $1/2$? One would certainly expect this to be true from normality/equidistribution-style properties, though those are...

Let $\pi_2(n)$ be the prime twin counting function. Let $li_2(x) = \int \frac{1}{\ln(t)}dx = li(x) - \frac{x}{\ln(x)}$ Let $C_2$ be the prime twin constant around $1.32$. Then we get the sharp conjecture for a prime $p > 5$ : $$|\pi_2(p) - C_2 li_2(p)| < \frac{\sqrt{p+2}}{3}+2$$ GENERALIZED...