Related papers: A Complete Answer to Erd\H{o}s Problem 690

In Pacific J. Math. 292 (2018), 223-238, Shareshian and Woodroofe asked if for every positive integer $n$ there exist primes $p$ and $q$ such that, for all integers $k$ with $1 \leq k \leq n-1$, the binomial coefficient $\binom{n}{k}$ is…

This paper establishes an extension of Wolstenholme's theorem to the ring of Gaussian integers $\mathbb{Z}[i]$. For a prime $p > 7$, we prove that the sum $S_p$ of inverses of Gaussian integers in the set $\{n+mi \mid 1 \leq n, m \leq p-1,…

Erd\H{o}s first showed that the number of positive integers up to $x$ which can be written as a product of two number less than $\sqrt{x}$ has zero density. Ford then found the correct order of growth of the set of all these integers. We…

We prove that any positive rational number is the sum of distinct unit fractions with denominators in $\{p-1 : p\textrm{ prime}\}$. The same conclusion holds for the set $\{p-h : p\textrm{ prime}\}$ for any $h\in\mathbb{Z}\backslash\{0\}$,…

We prove that for a positive integer $k$ the primes in certain kinds of intervals can not distribute too 'uniformly' among the reduced residue classes modulo $k$. Hereby, we prove a generalization of a conjecture of Recaman and establish…

Let $a$ and $b$ be positive integers. In 1946, Erd\H{o}s and Niven proved that there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1/b, 1/(a+b),..., 1/(an-a+b)$ are integers. In…

The following result, a consequence of Dumas criterion for irreducibility of polynomials over integers, is generally proved using the notion of Newton diagram: Let $f(x)$ be a polynomial with integer coefficients and $k$ be a positive…

Let P(x,d,a) denote the number of primes p<=x with p=a(mod d). Chebyshev's bias is the phenomenon that `more often' P(x;d,n)>P(x;d,r) than the other way around, where n is a quadratic non-residue mod d and r is a quadratic residue mod d. If…

The Erd\H{o}s-Straus conjecture, proposed in 1948 by Paul Erd\H{o}s and Ernst G. Straus, asks whether the Diophantine equation \[ \frac{4}{a} = \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \] admits positive integer solutions $b, c, d \in…

For a prime $p$, let $c(p)$ denote the limiting fraction of $n\times n$ matrices over $\mathbb{F}_p$ whose determinant is a primitive root modulo $p$. The quantity $c(p)$ is a natural multiplicative deformation of the totient ratio…

Let $I_k = [(2k-1)^2, (2k+1)^2)$ for $k \geq 1$. Starting from the odd-composite matrix $(b_{ij})$ with $b_{ij} = (2i-1)(2j-1)$, introduced by the author in [1], we define for each odd integer $n$ the \emph{matrix multiplicity} $r(n)$, the…

Let H(k) be the smallest N such that every finite coloring of [N] contains a monochromatic or rainbow k-term arithmetic progression. Erd\H{o}s and Graham asked whether $H(k)^{1/k}/k \to \infty$ (Problem #190 of the Erd\H{o}s Problems…

Motivated by problems of comparative genomics and paleogenomics, in [Chauve et al., 2009], the authors introduced the Gapped Consecutive-Ones Property Problem (k,delta)-C1P: given a binary matrix M and two integers k and delta, can the…

If $p\geq 5$ is prime and $k\geq 4$ is an even integer with $(p-1)\nmid k$ we consider the Eisenstein series $G_k$ on $\operatorname{SL}_2(\mathbb{Z})$ modulo powers of $p$. It is classically known that for such $k$ we have $G_k\equiv…

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…

Let K/Q be Galois, and let eta in K* whose conjugates are multiplicatively independent. For a prime p, unramified, prime to eta, let np be the residue degree of p and gp the number of P I p, then let o\_P(eta) and o\_p(eta) be the orders of…

Let $k\geq 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$ p_{n+1}-p_n\geq c\:\frac{\log p_n\: \log_2 p_n\: \log_4 p_n}{\log_3 p_n}\:,$$ with $c$ being a…

Let $v(k)$ be the smallest integer larger than $1$ that does not occur among the denominators in any identity of the form $$ 1=\frac1{n_1}+\cdots+\frac1{n_k}, $$ where $1 \le n_1<\cdots<n_k$ are pairwise distinct integers. In their 1980…

If $p_k$ is the k-th prime, the Firoozbakht conjecture states that the sequence $(p_k)^{1/k}$ is strictly decreasing. We use the table of first-occurrence prime gaps in combination with known bounds for the prime-counting function to verify…

If $a$ and $d$ are relatively prime, we refer to the set of integers congruent to $a$ mod $d$ as an `eligible' arithmetic progression. A theorem of Dirichlet says that every eligible arithmetic progression contains infinitely many primes;…