I am working through Poncelet Porisms and Beyond by V. Dragović and M. Radnović. In Chapter 3 (Section 3.2, "Algebraic curves in the complex projective plane"), I am stuck on an exercise that makes two claims about the intersection of lines with an algebraic curve. Here is the exact quote from...

This question was asked to me by a junior trying number theory problems and I got struck on it. Question:Show that there are infinitely many triples $(x,y,z)$ of integers such that $x^3+y^4=z^{31}$. I found a solution here https://www.isical.ac.in/~rmo/papers/rmo/rmo-2015-3.pdf : as problem $3$...

Background: I was exploring whether the standard normalization $\sum p_i = 1$ in probability is exact when the parts are written as real numbers. This led me to the quantity: $$\epsilon_n = \frac{1}{n}\left(1 - n^{-1/n}\right)$$ which represents a "gap" when dividing $1$ into $n$ equal parts....

The following is a rephrasing of Hartshorne Chapter II Exercise 2.3: For any ring $A$, let $A_{\text{red}}$ be the quotient of $A$ by its ideal of nilpotents. If $\mathcal{S}$ is a sheaf of rings on a topological space, let $\mathcal{S}_\text{red}$ be the sheafification of the presheaf $U...

Consider the reduced residue system modulo a primorial $$P_k=p_1p_2\cdots p_k.$$ Let $$R_k=\{r\in\{1,\ldots,P_k\}:\gcd(r,P_k)=1\},$$ ordered cyclically, and let $G_k$ be the cyclic sequence of gaps between consecutive elements of $R_k$. For example, for $$P_k=30=2\cdot3\cdot5,$$ the reduced...

In the following, I am going to compare two correspondences: $\textbf{correspondence between intermediate fields of $L/K$ and subgroups of ${\rm Aut}_K(L)$}$ vs $\textbf{correspondence between ideals of $R=k[x_1,\cdots,x_n]$ and algebraic subsets of $\mathbb{A}^n_k$}.$ While doing this, I will...

I was wondering about imaginary quadratic rings. Let the class number of the ring $R(p)$ be $c>1$. and let the norm be $N(x) = a^2 + b^2 p$ where $p$ is a prime. Let $f(x)$ be the number of factorizations into irreducibles of $x$. Let $N(x) > N(y)$ and let $g(N(x))$ be the largest value in...

I am a class 10 student exploring number theory and thinking about variations of famous problems like Legendre's conjecture.Let an interval between consecutive squares be defined for a given integer $n \ge 1$. Let $a_n$ represent the exact count of prime numbers that fall within the interval...

The action of $\mathrm{SL}_2(\mathbb{Z})$ on $\mathbb{H}$ (the upper-half plane) is defined by the Möbius transformation $$\gamma\cdot z\longrightarrow \frac{az+b}{cz+d}\qquad\text{where}\qquad\gamma=\begin{pmatrix} a & b \\ c& d \end{pmatrix}\in\mathrm{SL}_2(\mathbb{Z}).$$ I found that $\gamma...

In 2008, Jacobi-Madden found (essentially by data-mining the 25 smallest solutions) that $$a^4+b^4+c^4+d^4 = (a+b+c+d)^4 =e^4$$ was solvable and in fact a member of an infinite family. In early August 2026, Matej Veselovac and I were data-mining the first 45,000 smallest solutions of Eugene Go's...

I am self studying Algebraic geometry from Clader and Ross Beginning in Algebraic Geometry and I am struck on Proposition $9.50$ on page $270$. Background: For each $i \in ${$0,1...,n$}, the $i$ th affine patch of $\mathbb{P}^n$ is the set $\mathbb{A}_i^n=${$[a_0,a_1,...,a_n] \in...

Let $A$ be a commutative ring and $M$ an $A$-module. Suppose $M$ is free of rank $n$ with basis $\{ x_1, \ldots , x_n \}$. The $A$-module homomorphism $f \colon M \to A[X_1, \ldots, X_n]$ given by $x_i \mapsto X_i$ induces a unique $A$-algebra homomorphism $F \colon \operatorname{TS}_A(M) \to...

For any positive integer $n$, define $S(n)$ as the sum of the digits of $n$ in decimal (e.g., $S(19) = 1 + 9 = 10$). Define the sequence $a_k$ as: $a_1 = 7$ $a_{k+1} = a_k + S(a_k)$ Edit: To add, the first term ($a_1$) can be any positive integer(except $a_1 \not\equiv 0 \pmod 3$). I conjecture:...

I am self studying Algebraic Geometry from the textbook of Clader and Ross and I have question in Propositions $8.4$ on page $217-218$. Proposition $8.4$: Let $X\subseteq \mathbb{A}^m $ and $Y\subseteq \mathbb{A}^n$ be affine varieties. Then we have $I(X\times Y)= \langle I(X)\rangle+\langle...

I have been self studying the textbook: Beginning in Algebraic Geometry by Clader and Ross and found this book very wonderful simply because most of the proofs are proved in the textbook itself( I donot have a help in real life) and it has a good number of exercises. A lot of intuition is also...

I am curious about infinite series and definite integrals for which the closed-form evaluation is currently an open problem. Now, one can of course think of some strange, complicated infinite series like $$ S:= \sum_{n=1}^{\infty} \frac{1}{n^{n!}} \ \ , $$ but I am not interested in such series....

This might be something trivial I missed but I was wondering. intro Let $w(n)$ be the number of distinct prime factors of the positive integer $n$. We know that $w(n)$ grows slowly. It seems logical to me that $$ \max(i<n,w(i))=A$$ $$\to w(n) < A+2$$ The simpler analogue conjecture for the count...

A prime number $p$ is called a "Wieferich prime number to base $q$" or "$q$-ary Wieferich prime number" granted $$p^2\;\vert\;q^{p-1}-1$$ and hence by a "Wieferich prime number" here it is meant a binary Wieferich prime number. The only known Wieferich prime numbers are $1093$ and $3511$ but the...

Let $p$ be a prime. Experimental data shows that for any $1 \le m < p$, $$ \lim_{p \to \infty} \left(\sum_{\substack{1\le a,b<p\\ab\equiv m\pmod p}} \frac{1}{b}\left(\frac{b}{a}\right)^{1/p} - \log p \right) = \gamma. $$ The result is not true if $p$ is composite. Can this be proved for...

I have been self studying Algebraic Geometry from Clader and Ross's Beginning in Algebraic Geometry and I have a question on Page $195$: Proposition $7.8$. Background information: If $f \in K[x_1,...,x_n]$ and $a=(a_1,...,a_n)\in \mathbb{A}^n$ , then the linearization of $f$ at $a$ is defined by...