For even number $n$, if $n$ is not a totient (i.e. not in the range of Euler totient function), then $n$ is not in OEIS sequence A252502, but if $n$ is a totient, must be $n$ in A252502? For odd number $n$, if $n-1$ is not a totient (i.e. not in the range of Euler totient function), then $n$ is...

Erdős Problem 252:(https://www.erdosproblems.com/252) Let $k\geqslant 1$ and $\sigma_{k}(n):= \sum_{d|n} d^{k}$. Is $\displaystyle \sum_{n=1}^{\infty} \frac{\sigma_{k}(n)}{n!}$ irrational? Currently this problem is open for $k \geqslant 5$. I am considering a different version of this problem...

I know that Schur and Murty proved that an Euclidean proof (hence a "non-analytic" proof) for the existence of infinite primes $p \equiv \ell \mod q$ with $q$ and $\ell$ coprime can be given if and only if $\ell^2 \equiv 1 \mod q$. Are there any cases where $\ell^2 \not\equiv 1 \mod q$, but we...

In this related question, @Gerry Myerson asked if the conjecture fails for composites. Upon examining for composites, I found a pattern for module $n^2$ instead of $n^3$ in the linked original question. I have verified these conjectures for $n \le 5 \times 10^6$. Can they be proved or disproved....

Let $F_1,F_2$ be degree $1$ and $F_3$ be degree $2$ in $\mathbb Z[x_1,\dots,x_4]$. Let there be an unique common integer to $F_i$. Let them be algebraically independent of a fourth polynomial $G$ which also has the same common integer root. Is it possible for the system to not form a complete...

I had posted this question in MO that has remained unanswered in MO for more than two years now. While working on it, I accidently found an unexpected result. Let $0<x\leq y\leq z$ be the ordered side lengths of the triangle determined by three independent uniformly distributed points on a...

It is a classical result that the roots of a general cubic polynomial $x^3 + ax^2 + bx + c = 0$ can be expressed via Cardano’s formula using radicals of the form: $$x=\sqrt[3]u+\sqrt[3]v+k$$ where $u$ and $v$ depend on the coefficients and the discriminant $\Delta$. ​I am curious about the...

My experimental data suggests that the following binomial sum congruence involving primes and Fibonacci numbers hold. I have experimentally verified them for all primes $\le 1.5 \times 10^6$. Can they be proved or disproved? Conjecture 1. Let $p$ be a prime, $ p\equiv 1 \text{ or } 19 \pmod{30}....

I. Question Consider the following independent and symmetric mixed equations, $\quad x_1^8+x_2^8+x_3^4 = y_1^8+y_2^8+y_3^4$ $\quad x_1^8+x_2^8+x_3^4+x_4^4 = y_1^8+y_2^8+y_3^4+y_4^4$ $\quad x_1^8+x_2^8+x_3^8+x_4^4 = y_1^8+y_2^8+y_3^8+y_4^4$ $\quad x_1^8+x_2^8+x_3^8+x_4^8+x_5^4 =...

Let $$ 0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0 $$ be a short exact sequence of vector bundles. It is well known that for exterior powers there exists a natural filtration $$ 0=F_{r+1}\subset F_r\subset \cdots \subset F_0=\bigwedge^r B $$ such that $$...

I am investigating the Diophantine equation $$x^4 - Ny^4 = 1$$ where $N$ can be expressed as a sum of three fourth powers: $$N = a^4 + b^4 + c^4$$ for integers $a, b, c$. Background: I see some from $\mathrm{A356496}$ for squarefree integers $k$ such that $x^4 - ky^2 = 1$ has a nontrivial...

Title: Primes as $X-Y$ with $\gcd(X,Y)=1$ and $\operatorname{rad}(XY)$ equal to the product of all primes below $p$ — is this known? While experimenting with multiplicative decompositions of primes, I arrived at the following question. I would like to know whether it is already in the...

Consider a non-cuspidal point $x\in X_0(p)(K)$ representing an elliptic curve $E/K$ over a quadratic field $K$, and suppose that $E$ has potentially multiplicative reduction over a prime $\mathfrak{q}$ of $K$ over the rational prime $q$. Let $0,\infty$ be the cusps of $X_0(p)$. Using the...

This is about the umpteenth discrepancy of definitions in algebraic geometry. It's about recipes to define properties of morphisms of algebraic spaces out of properties of morphisms of schemes. When the morphism of algebraic spaces in question is representable the recipe is clear [SP, 025V;...

We say that a topological space $X$ is irreducible if it can not be written as the union of two proper closed subsets. My question is: Do we consider this definition when the topological space is defined in terms of open subsets or closed subsets? When working with the zariski topology, the...

I am looking for a topological or geometric interpretation of the Albert–Brauer–Hasse–Noether (ABHN) theorem, which establishes the local-global principle for central simple algebras over a global field $K$. The classical exact sequence is given by: $$0\rightarrow \text{Br}(K)\rightarrow...

We consider the $L$-function associated with a nonzero Maass cusp form $f$ of weight $0$, level $N$, and Laplace eigenvalue $1/4+r^2$. Let $t(n)$ be the normalized Fourier coefficient corresponding to the Maass cusp form $f$. I need the estimate of $$\sum_{n\le T}t(n)e^{2\pi i n x},$$ where $x...

One question from a past qualifying exam is to find the Galois group of $x^6+3$ over the finite field $\mathbb{F}_5$. With some creativity, we can write: $$x^6+3=x^6+8=(x^2)^3+2^3=(x^2+2)(x^4-2x^2+4)=(x^2+2)(x^4+4x^2+4-16x^2)\\ =(x^2+2)((x^2+2)^2-(4x)^2)=(x^2+2)(x^2-2x+2)(x^2+2x+2).$$ Since...

One question in a past qualifying exam asked us to find the Galois group of $x^6+3$ over $\mathbb{F}_5$, $\mathbb{F}_7$, and $\mathbb{Q}$. Each part of this question has been answered individually on this website: over $\mathbb{F}_5$, $\mathbb{F}_7$, and $\mathbb{Q}$. I asked this question again...

I am reading Arithmetic Moduli of Elliptic Curves by Katz and Mazur, and I have a question about the beginning of Chapter 2, §2. Let $f:E\to S$ be an elliptic curve. Since the sheaf of relative diffrentials $\Omega^1_{E/S}$ is an invertible sheaf on $E$, one defines...