Related papers: Subproducts of small residue classes

If p is a prime, then the numbers 1, 2, ..., p-1 form a group under multiplication modulo p. A number g that generates this group is called a primitive root of p; i.e., g is such that every number between 1 and p-1 can be written as a power…

We prove that, for every modulus $\mathfrak{q}$, every class of the narrow ray class group $H_{\mathfrak{q}}(\mathbf{K})$ of an arbitrary number field $\mathbf{K}$ contains a product of three unramified prime ideals $\mathfrak{p}$ of degree…

In this paper we give simple expressions, involving binomials coefficients, for the value of $c(n,k)$ modulo $p^{v_p(n)}$, when $v_p(n) > 0$. Here $c(n,k)$ denotes a Stirling number of the first kind, and $v_p(n)$ is the highest power of…

For an odd prime $p$, we say $f(X) \in {\mathbb F}_p[X]$ computes square roots in $\mathbb F_p$ if, for all nonzero perfect squares $a \in \mathbb F_p$, we have $f(a)^2 = a$. When $p \equiv 3 \mod 4$, it is well known that $f(X) =…

Let $p$ be a prime number and $\Bbbk=\bar{\mathbb{F}}_p$, the algebraic closure of the finite field $\mathbb{F}_p$ of $p$ elements. Let ${\bf G}$ be a connected reductive group defined over $\mathbb{F}_p$ and ${\bf B}$ be a Borel subgroup…

A group in which every element commutes with its endomorphic images is called an $E$-group. If $p$ is a prime number, a $p$-group $G$ which is an $E$-group is called a $pE$-group. Every abelian group is obviously an $E$-group. We prove that…

Let $E$ be an elliptic curve defined over ${\mathbb Q}$. For a prime $p$ of good reduction for $E$, denote by $e_p$ the exponent of the reduction of $E$ modulo $p$. Under GRH, we prove that there is a constant $C_E\in (0, 1)$ such that $$…

We study the existence of products of primes in arithmetic progressions, building on the work of Ramar\'e and Walker. One of our main results is that if $q$ is a large modulus, then any invertible residue class mod $q$ contains a product of…

Issai Schur once asked if it was possible to determine a bound, preferably using elementary methods, such that for all prime numbers p greater than the bound, the greatest number of consecutive quadratic non-residues modulo p is always less…

The minimal excludant of a partition $\lambda$, $\rm{mex}(\lambda)$, is the smallest positive integer that is not a part of $\lambda$. For a positive integer $n$, $ \sigma\, \rm{mex}(n)$ denotes the sum of the minimal excludants of all…

Let $p$ be a prime number and let $S=\{x^p+c_1,\dots,x^p+c_r\}$ be a finite set of unicritical polynomials for some $c_1,\dots,c_r\in\mathbb{Z}$. Moreover, assume that $S$ contains at least one irreducible polynomial over $\mathbb{Q}$. Then…

Let $N$ be a finite set, let $p \in (0,1)$, and let $N_p$ denote a random binomial subset of $N$ where every element of $N$ is taken to belong to the subset independently with probability $p$ . This defines a product measure $\mu_p$ on the…

Two conjectures, posed by Finch-Smith, Harrington, and Wong in a paper published in Integers in $2023$, are proven. Given a monic biquadratic polynomial $f(x) = x^4 + cx^2 + e$, we prove a formula for the sum of its distinct outputs modulo…

Following the natural instinct that when a group operates on a number field then every term in the class number formula should factorize `compatibly' according to the representation theory (both complex and modular) of the group, we are led…

We prove the conjecture by M. Yip stating that counting genus one partitions by the number of their elements and parts yields, up to a shift of indices, the same array of numbers as counting genus one rooted hypermonopoles. Our proof…

Let $D$ be an open disk of radius $\le 1$ in $\mathbb C$, and let $(\epsilon_n)$ be a sequence of $\pm 1$. We prove that for every analytic function $f: D \to \mathbb C$ without zeros in $D$, there exists a unique sequence $(\alpha_n)$ of…

Let $E$ be an elliptic curve defined over $\mathbb Q$ and $\widetilde{E}_p$ denote the reduction of $E$ modulo a prime $p$ of good reduction for $E$. The divisibility of $|\widetilde{E}_{p}(\mathbb{F}_p)|$ by an integer $m\ge 2$ for a set…

In this paper we establish some new supercongruences motivated by the well-known fact $\lim_{n\to\infty}(1+1/n)^n=e$. Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{-1/(p+1)}k^{p+1}\equiv 0\ \pmod{p^5}\ \ \ \mbox{and}\ \ \…

In this manuscript, a solution to Problem 18.91(b) in the Kourovka Notebook is given by proving the following theorem. Let $P$ be a Sylow $p$-subgroup of a group $G$ with $|P| = p^n$. Suppose that there is an integer $k$ such that $1 < k <…

Let $\mathbb{F}_p$ be the field of a prime order $p.$ It is known that for any integer $N\in [1,p]$ one can construct a subset $A\subset\mathbb{F}_p$ with $|A|= N$ such that $$ \max\{|A+A|, |AA|\}\ll p^{1/2}|A|^{1/2}. $$ In the present…