Let $k$ be an algebraically closed field and $V$ be an affine variety. From Page 19 of Daniel Perrin’s Algebraic geometry. Lemma: Let $F,G\in k [X,Y]$ be non-zero polynomials without common factors, there is a non-zero polynomial $d\in k[X]$ and polynomials $A,B\in k[X,Y]$ such that $d= AF+BG$...

I am self studying Algebraic Geometry from Daniel Perrin's Algebraic Geometry. $k$ is an algebraically closed field. $\Gamma(V)$ is image of $r: k[X_1,...,X_n]\to F(V,k)$ and $V,W$ is affine algebraic set. $\theta : \Gamma(W) \to \Gamma(V)$ be a homomorphism of $k-$algebras and $\phi: V\to k^m$...

I am self studying Algebraic Geometry from the Daniel Perrin's textbook. $k$ is an algebraically closed field. $\Gamma(V)$ is image of $r: k[X_1,...,X_n]\to F(V,k)$ and $V,W$ is affine algebraic set. $\theta : \Gamma(W) \to \Gamma(V)$ be a homomorphism of $k-$algebras and $\phi: V\to k^m$ is the...

Let $$B:=\{n\geq 13: n \text{ odd and there exists a prime } q \text{ such that } 9n^2+1-q \text{ is prime, where either } q=3, \text{or } q\geq 11,q \equiv 2\pmod 3, q-1 \text{ is not a square}\}$$ I am trying to understand whether the following “almost-all Goldbach” statement is known /...

I am exploring a family of integer sequences $(k_n)$ that combine explosive quadratic growth with specific "modular" reduction rules based on the proximity to perfect squares and powers of two. I’ve categorized these reduction mechanisms as "Valves." The Core Growth Function: Let $f(x) = ax^2 +...

Let $(a,b,c)$ be positive integers such that a $abc+1 \mid a^2+b^2+c^2$. Then $\dfrac{a^2+b^2+c^2}{abc+1}$ can be written as the sum of 2 positive squares. Proposed by Sam Vandervelde.

Professor Manin's 1990 ICM talk states there is a convincing case to be made that these groups are, in some sense, "more fundamental" in number theory than even the integers. The talk's purpose being a broadest-possible survey of the works of Professor Drinfel'd, Professor Manin does not deem it...

$k$ is an algebraically closed field. $\Gamma(V)$ is image of $r: k[X_1,...,X_n]\to F(V,k)$ and $V$ is affine algebraic set. I am self studying Algebraic Geometry from Daniel Perrin's Algebraic Geometry and have a question in proof of Proposition $4.8 $ of Chapter $1$ on Page $17$. Proposition...

Let $$ t_n = {16 \over (2n+1)5^{2n+1}} - {4 \over (2n+1)239^{2n+1}}. $$ Define $$ S_N = \sum_{n=0}^N (-1)^n t_n. $$ For each integer $m \geq 0$, define $$ I_m = [S_{2m+1}, S_{2m}]. $$ By Machin's formula, $$ \pi = 16\arctan(1/5) - 4\arctan(1/239), $$ so $$ S_{2m+1} \leq \pi \leq S_{2m}. $$ Also,...

This is extended computational evidence for the Conjecture from this question, focusing on the $n=1$ case. Conjecture A for $n=1$ states: for all sufficiently large primes $p$ with $q \mid p-1$, $$v_p\!\left(\prod_{k=1}^{p} \Phi_q(k)\right) = q-1.$$ Reduction to $v_p(k^{p-1}-1)$. By the...

Let $N_n=\binom{2n}{n}^3(42n+5)$. Define $L_m=\sum_{n=0}^m \frac{N_n}{2^{12n+4}}$ and $U_m=L_m+\frac{4N_{m+1}}{3\cdot 2^{12m+16}}$. These are rational intervals coming from a Ramanujan series for $1/\pi$. For this question, I only want to study rational points inside these explicit intervals....

In Kedlayas lecture notes (https://kskedlaya.org/prismatic/sec_overview.html) about prismatic cohomology he introduces the ring of Witt vectors using $\delta$-rings via the fact that the Witt vector functor $W$ is the right adjoint to the forgetful functor $\mathbf{Ring}_{\delta}\to...

Let $G$ be a group object in an $\infty$-topos $\mathcal T$, and let $P \to X$ be a $G$-torsor as in the accepted answer by Daniël Apol to this question; equivalently, as in the answer, such a $G$-torsor is given by a map $\varphi \colon X \to BG$, and $P$ is the pullback of $\varphi$ along the...

Let us consider a prime number $p$ and three distinct positive integers $n_1<n_2<n_3$ less than $p$. Let us assume that the triple $(n_1, n_2, n_3)$ satisfies the following condition $$k+[kn_1]+[kn_2]+[kn_3]=2p, \ \ for \ all \ 1\leq k\leq p-1$$ Where $[kn_i]$ denotes the rest of the division of...

For a semiprime $N = p×q$, consider the sequence $f(k) = k^2 + k + N$, for $k = 0, 1, 2, ...$ Since $f(k) ≡ k(k+1) \mod N$, we have gcd($f(k), N$) = gcd($k(k+1), N$). This means a factor of $N$ is revealed at position $k$ whenever $p$ divides $k$ or $k+1$. For $N = 77 = 7×11, f(k) = k^2 + k +...

While trying to prove FLT for $n=7$, I came to find that there could be a certain class of solutions (counterfactual) for the counterfactual case that FLT is false for $n=7$: $(d^7+x^7+y^7)^7 = (d^7+x^7-y^7)^7 + (d^7-x^7+y^7)^7$ where $x, y$ and $d$ are coprime and all of them are coprime to...

For an elliptic curve, an L-function is associated and has an Euler product. Is the Euler factor at prime $p$ the same as the zeta function of this elliptic curve over $\mathbb{F}_p$ at $p^{-s}$? That is $\exp(\sum_{n=1}^{\infty}\frac{N_n}{n}t^n)$ with $t=p^{-s}$, where $N_n$ is the number of...

Let $\varphi: X\to Y$ be a dominant morphism of affine varieties over algebraically closed field $k$, $\varphi*: k[Y]\to k[X]$ be the induced $k$-algebra monomorphism. Let $\mathfrak{m}_y$ be the ideal of a point $y\in Y$. I believe the ideal $I(\varphi^{-1}(y))$ of the fiber $\varphi^{-1}(y)$...

Let $G$ be a finite group, let $k$ be a field of characteristic $p$, and let $S_p$ be a Sylow $p$-subgroup of $G$. It is a well-known fact from modular representation theory that every irreducible representation factors through any normal $p$-subgroup of $G$. In particular, it factors through...

Let $\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D}, \mathcal{E}, \mathcal{F}$ be infinite arithmetic progressions of integers. We consider the set of all second-order quadratic recurrence relations: $$s_{n+1} = a s_n^2 + b s_n s_{n-1} + c s_{n-1}^2 + d s_n + e s_{n-1} + f$$ where $(a, b, c,...