I am self studying algebraic Geometry from the textbook of Clader and Ross Beginning in Algebraic Geometry and I have $2$ in the proof of the theorem $7.18$ given on the page $201-203$. Background: define $I_a= \{F\in K[X]\mid F(a)=0\} \subseteq K[X]$. On page $201$ we see that $I_a$ can be...

I was self studying Algebraic Geometry from the textbook of Algebraic Geometry by Clader and Ross: Beginning in Algebraic Geometry. I have questions on page $206-207$ of the textbook. Background information: If $f \in K[x_1,...,x_n]$ and $a=(a_1,...,a_n)\in \mathbb{A}^n$ , then the linearization...

Let the sequence $(a_n)$ be defined by $$ a_1=3,\qquad a_{n+1}=a_n^2-2 $$ for every positive integer $n$. Let $p_n$ be the smallest prime divisor of $a_n$. Prove that $$ p_n\ge 2n+3 $$ for every $n\ge 2$. Using mathematical induction, we prove that $ a_n=\alpha^{2^{n-1}}+\beta^{2^{n-1}} $ for...

Prove that there are no positive integers $a$, $b$, and $c$ that satisfy the following equation: $$a^2 + b^2 = 3c^2.$$ Think about remainders modulo $3$ (Modular Arithmetic $\bmod 3$). What are the possible remainders of any perfect square when divided by $3$? Then, apply the method of Infinite Descent.

Is this identity (13), which links the Goldbach $\varphi$ function to the local count of partitions, known? https://zenodo.org/records/22048675

I am reading "The Rising Sea" by Ravi Vakil. I have several question regarding section 3.2 . We know $\operatorname{Spec}\mathbb{C}[x]$ looks like, and we can associate each maximal ideal to a complex number. But what about genric point? What is the intuition behind the terminology: the elements...

Let $E$ be a vector bundle of rank $2$ on $\mathbb P^2$ and $Z$ be zero dimensional subscheme of length $k$ supported on a point $p \in \mathbb P^2$. What is the number $h^0(E(m) \times \mathscr O_Z)$ for an integer $m$? My intuition is that it should be $2k$. But it seems in some literature it...

Let $S=k[x_0,x_1;y_0,y_1]$ be the bihomogeneous coordinate ring of $\mathbb{P}^1\times\mathbb{P}^1$. Suppose that $F\in S_{(d_1,d_2)}$ is a generic bihomogeneous form of bidegree $(d_1,d_2)$, and let $F_1\in S_{(a_1,b_1)}$, $F_2\in S_{(a_2,b_2)}$ be generic forms of lower bidegrees. The question...

I'm looking at the family of step functions $\Phi_p(x) = \pi(2x - p)$, where $\pi$ is the prime-counting function and $p$ ranges over the primes. When I superimpose these functions for many values of $p$ (see attached figure, $n$ up to $5 \times 10^7$), the result is a dense region with internal...

Let $m_1,\ldots,m_s$ be pairwise coprime positive integers, and let $$ M=\prod_{i=1}^s m_i. $$ For each $i$, let $A_i$ be a non-empty proper subset of residue classes modulo $m_i$. Define the CRT-allowed residue set $S \pmod M$ by $$ S=\{x \pmod M : x \pmod {m_i} \in A_i \text{ for every } i\}....

I found the following infinite continued fraction: $$ \operatorname{C_{B4CC}}=\cfrac{2}{1+\cfrac{2}{1+\cfrac{6}{1+\cfrac{3}{1+\cfrac{10}{1+\cfrac{4}{\ddots}}}}}} $$ where $\operatorname{C_{B4CC}}$ denotes Baxter's four-coloring constant and the partial numerators are defined by interweaving...

I am studying a deterministic construction based on the consecutive gaps between primes. The phenomenon I am interested in is NOT that arithmetic on primes sometimes produces another prime. The interesting point is that the same future value can be generated by different rows of a cumulative...

Let $x,n$ be positive integers. Consider the equation $$e^x+x^e=e^n$$ I would like to know whether this equation can have any positive integer solutions. Since $x^e \gt 0$, any solution must have $n \gt x$. Dividing by $e^x$, $$1+\left(\frac{x}{e}\right)^e=e^{n-x},$$ so $$n-x=...

I'm reading CLS's Toric Varieties right now, and something is confusing me. By Theorem 1.1.17, an affine variety $V$ is toric iff $I(V)$ is toric, i.e. prime and generated by binomials. Now the variety $V = V(X+Y-Z) \subset \mathbb{C}^3$ seems to me to be toric simply because it's isomorphic to...

I was playing around with Goldbach representations and came up with the following class of primes. Consider an even integer $n$ that can be written as a sum of two odd primes, $$ n=p+q,\qquad p\le q. $$ For a fixed $n$, define $GGT(n)$ to be the largest possible value of $q$ among all such pairs...

P.412 of volume 3 of Gauss's collected works contains two unpublished remarks of Gauss that apparently have not been discussed yet. The first one is of number-theoretic significance, while the second relates the value of $Q([a+bi]\varphi)$ at a point $\varphi=\text{arcsinlemn} (x)$ such that...

Let $p_n$ be the $n$-th prime and let $$ g_n = p_n-p_{n-1}. $$ Define $$ S_n=\sum_{i=2}^{n} g_i=p_n-2 $$ and $$ V_n=S_n+g_n=p_n+g_n-2. $$ I call a convergence the occurrence of the same value $V$ at two or more distinct positions. For example, for $V=103$: $$ 95+8=103,\qquad 99+4=103,\qquad...

I am seeking guidance on how to properly classify the singular projective surface $S \subset \mathbb{P}^3$ defined by the degree 4 homogeneous polynomial $y^4 - 4x^3w + 8w^4 - z^2w^2 = 0$, which is known to contain some elliptic curve of rank 1, alongside a unique isolated singularity at $P_0 =...

I'm studying theorem III.7.5 from Hartshorne's book. There are a few things I don't understand in this proof. Theorem. Let $X$ be a projective scheme over a field $k$. Then $X$ has a dualizing sheaf $\omega_X^\circ$. Proof. Let $\dim(X) = n$. Embed $X$ as a closed subscheme of $P:=\mathbb{P}^N$...

Let $p$ be an odd prime and let $n \ge 3$. I have been considering the following family of polynomials. Set $$ r_n:=\frac{1}{2\sqrt{n-1}}, $$ and define $$ B(p,n) := r_n+\frac{p+1}{r_n}+\frac{2p}{r_n^{n-1}}. $$ Let $$ A(p,n) := 2+2p\left( \left\lfloor \frac{B(p,n)-2}{2p} \right\rfloor+1 \right)....