For a commutative ring $R$ and an $R$-module $M$, let $\operatorname{Gr}(n,M)$ be the Grassmann functor from the category of $R$-algebras to sets. As I understand it, it should be possible to define a subfunctor $U$ when fixing $x_1, \ldots, x_n$ elements of $M$, how does this subfunctor look...

Let all terms be positive. There are infinitely many solutions to, $$a^2+b^2 = (b+1)^2$$ $$a^3+b^3+c^3 = (c+1)^3$$ like the well-known $3^2+4^2 = 5^2$ and $3^3+4^3+5^3 = 6^3$. However, this has versions for higher degrees. For $4$th powers by Jaroslaw Wroblewski, $$178^4 + 1345^4 + 10400^4 +...

(Updated with a computer search.) I. Question We seek to find infinitely many primitive solutions to, $$a^4+b^4+c^4 = d^4+e^4$$ where $d \color{red}{\ne} e$. The most well-known case when $d = e$ is, $$a^4+b^4+(a+b)^4 = 2(a^2+ab+b^2)^2$$ where one then solves $a^2+ab+b^2 = z^k$ for $k=2$. (In...

It is known that every finitely generated projective module $M$ over a commutative ring $A$ has locally constant rank, i.e., for each $\mathfrak{p} \in \mathrm{Spec}(A)$, there are non-negative integer $r$ and an open neighborhood $U \subseteq \mathrm{Spec}(A)$ such that, for every $\mathfrak{q}...

In the recent IMO 2026 Problem 1, a blackboard game is played where two active integers $m, n > 1$ are repeatedly selected and replaced by: $$g = \gcd(m, n) \quad \text{and} \quad l' = \frac{\operatorname{lcm}(m, n)}{\gcd(m, n)}$$ It is a known result that this game must terminate in a finite...

I have been studying the following digit dynamical system. Let $E(n)$ denote the sum of the even digits of $n$, and let $O(n)$ denote the sum of the odd digits of $n$. Define $$ f(n)=\big(E(n)\,O(n)\big)^2. $$ I am interested in the dynamical system obtained by repeated iteration $$...

Let $G=\operatorname{Spec}(R)$ be a finite flat commutative group scheme over $S=\operatorname{Spec}(A)$ of characteristic $p>0$. Suppose the $p$-Lie algebra $\operatorname{Lie}(G/S)$ is locally free. I would like to know the $p$-mapping on $\operatorname{Lie}(G/S)$ coincides with differential...

for the diophantine $x_1^8+x_2^8+x_3^8+x_4^8=y_1^8+y_2^8+y_3^8+y_4^8 $ Back in 2006, Nuutti Kuosa discovered the following non-trivial integer solution $$1953^8+2012^8+3113^8+861^8=1128^8+2767^8+2557^8+2823^8$$ A system of equations was used to speed up the search....

I am working on a sentence of Yves Hellegouarch in his book "Invitation aux mathématiques de Fermat-Wiles" . In the Fermat section, page 38, he tells that Fermat probably associated the equation $z^p=x^p+y^p$ to the form $X^2+(-1)^{(p+1)/2}pY^2$. I don't really understand the reason he thinks...

I am curious if there is a notion of how to find a Gröbner basis for any ideal $I$ of a finitely generated algebra $R\cong \mathbb{K}[x_1,\dots,x_k]/J$. I know that Gröbner bases are generaly developed as a tool for polynomial rings, but I wonder what fails in this case or in which cases it's...

I'm struggling to understand an argument in the book "Weil Conjectures, Perverse Sheaves, and $l$-adic Fourier Transform" by Kiehl and Weissauer. In Theorem I.6.1, they prove (in specific cases) that the $i$-th cohomology of a pure sheaf of weight $w$ has weight $w+i$. I'm confused by their...

I am trying to prove that the prime ideals above 5 in $K=\mathbb Q(\sqrt{359})$ are non-principal. I calculated the splitting to be $5O_K = (5,\sqrt{359} + 2)(5,\sqrt{359} + 3)$. If either of the ideals were principal their generator would have norm $\pm 5$. Showing that $a^2-359b^2 = -5$ has no...

Let $\mathcal P$ denote the set of prime numbers, and consider $$ N =\frac{1}{1-\sum_{p\in\mathcal P}2^{-p}}. $$ Numerically, the binary expansion of $N$ appears to behave like that of a base-$2$ normal number. For example, among the first $10^6$ binary digits, the frequencies of $0$ and $1$,...

Let $N \geq 5$ be an integer so that the $\Gamma_1(N)$-moduli problem is representable over $\mathbb{Z}[1/N]$ (both in terms of elliptic curves/generalized elliptic curves). I am interested in Katz modular forms of this level and their $q$-expansions. From my modest understanding, there are two...

As a soon-to-be senior undergraduate planning to pursue research in Schubert calculus under a supervisor specializing in this field, I have struggled to locate accessible introductory textbooks or lecture notes for this subject, as well as more advanced reference materials to save for my future...

There is a philosophy attributed to Grothendieck that it is better to have a good category (e.g. mapping objects, abelian category, etc) with bad objects than a bad category with nice objects. What are some examples of this? Please also describe some ways these good categories have been helpful.

Let $p \geq 3$ be an odd prime. I would like to know whether the following conjecture is true. Conjecture For every odd prime $p \geq 3$, there exist integers $a$ and $b$ such that: $a+b+3$ is prime; $4a+2b+3=p$ $\gcd(a,b,3)=1$ $b^2\leq 4a$ $a\geq 1$. Here, $\mathbb{P}$ denotes the set of prime...

In the rather new MSE post Recursive sequence $a_{k+1} = a_k - \gcd(a_k, (n+k)^2 - 1)$ generating twin primes, a sequence of integers $(a_k)_{k \geq 1}$ is associated to each natural number $n$, namely $a_0 = n^2$ and $a_{k+1} = a_k - \gcd(a_k,(n+k)^2-1)$. What is interesting, as pointed out by...

Let $X$ be a smooth algebraic variety over a field $k$, and let $K/k$ be any field extension. I want to prove that $$ X_K:=X\times_{\operatorname{Spec}k}\operatorname{Spec}K $$ is smooth over $K$. I want to use only the following facts: Jacobian criterion (rational points): If $$...

I fail to understand the highlighted statement in my screenshot below. If $U \subset Y$ is a non empty open subset, then the natural map $g: \mathscr{O}_Y(U) \to k(Y)$ given by $(U,f) \mapsto [U,f]$ is naturally injective. Indeed, if $g((U,f_1)) = g((U,f_2))$, i.e $[U, f_1] = [U,f_2]$, then...