Our objective is to provide an upper bound for the length $\ell_N$ of the longest run of consecutive integers smaller than $N$ which have the same number of divisors. We prove in an elementary way that $\log\ell_N\ll(\log N\log\log N)^λ$, where $λ=1/2$. Using estimates for the Jacobsthal function, we then improve the result to $λ=1/3$.
We establish new upper bounds for the length of runs of consecutive positive integers each with exactly $k$ divisors, where $k$ is a given positive integer of some special forms. Also we have found exact values of the maximum possible runs for some fixed values of $k$. In addition, we exhibit the run of 11 consecutive positive integers each with exactly 36 divisors, the run of 13 consecutive positive integers each with exactly 12 divisors, the run of 17 consecutive positive integers each with exactly 24 divisors, and the run of 17 consecutive positive integers each with exactly 48 divisors.
We have found maximum possible runs of consecutive positive integers each having exactly $k$ divisors for some fixed values of $k$. In addition, we exhibit the run of 10 consecutive positive integers each having exactly 12 divisors and two runs of 14 consecutive positive integers each having exactly 24 divisors.
In a previous paper, the authors proved that in any system of three linear forms satisfying obvious necessary local conditions, there are at least two forms that infinitely often assume $E_2$-values; i.e., values that are products of exactly two primes. We use that result to prove that there are inifinitely many integers $x$ that simultaneously satisfy $$ω(x)=ω(x+1)=4, Ω(x)=Ω(x+1)=5, \text{and} d(x)=d(x+1)=24.$$ Here, $ω(x), Ω(x), d(x)$ represent the number of prime divisors of $x$, the number of prime power divisors of $x$, and the number of divisors of $x$, respectively. We also prove simila…
We show that for integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the normal order of the number of prime factors is $C \log \log n$, where $C=(1-e^{-γ})^{-1} = 2.280...$ and $γ$ is Euler's constant. We explore several applications and resolve a conjecture of Margenstern about practical numbers.
An integer $n$ is called practical if every $m\le n$ can be written as a sum of distinct divisors of $n$. We show that the number of practical numbers below $x$ is asymptotic to $c x/\log x$, as conjectured by Margenstern. We also give an asymptotic estimate for the number of integers below $x$ whose maximum ratio of consecutive divisors is at most $t$, valid uniformly for $t\ge 2$.
We say a polynomial f having integer coefficients is strongly coefficient convex if the set of coefficients of f consists of consecutive integers only. We establish various results suggesting that the divisors of x^n-1 with integer coefficients have the tendency to be strongly coefficient convex and have small coefficients. The case where n=p^2*q with p and q primes is studied in detail.
Let e>=1 and b>=2 be integers. For a positive integer n=\sum_{j=0}^ka_jb^j with 0<=a_j<b, define T_{e,b}(n)=\sum_{j=0}^ka_j^e. n is called (e,b)-happy if T_{e,b}^r(n)=1 for some r>=0, where T_{e,b}^r is the r-th iteration of T_{e,b}. In this paper, we prove that there exist arbitrarily long sequences of consecutive (e,b)-happy numbers provided that e-1 is not divisible by p-1 for any prime divisor p of b-1.
A positive integer $n$ is called an abundant number if $σ(n)\ge 2n$, where $σ(n)$ is the sum of all positive divisors of $n$. Let $E(x)$ be the largest number of consecutive abundant numbers not exceeding $x$. In 1935, P. Erd\H os proved that there are two positive constants $c_1$ and $c_2$ such that $c_1\log\log\log x\le E(x)\le c_2\log\log\log x$. In this paper, we resolve this old problem by proving that, $E(x)/\log \log\log x$ tends to a limit as $x\to +\infty$, and the limit value has an explicit form which is between $3$ and $4$.
In this work, we discuss a number game that develops in a manner similar to that on which Gilbreath's conjecture on iterated absolute differences between consecutive primes is formulated. In our case the action occurs at the exponent level and there, the evolution is reminiscent of that in a final Ducci game. We present features of the whole field of the game created by the successive generations, prove an analog of Gilbreath's conjecture, and raise some open questions.
Let $n$ be a positive integer and $f(x) := x^{2^n}+1$. In this paper, we study orders of primes dividing products of the form $P_{m,n}:=f(1)f(2)\cdots f(m)$. We prove that if $m > \max\{10^{12},4^{n+1}\}$, then there exists a prime divisor $p$ of $P_{m,n}$ such that ord$_{p}(P_{m,n} )\leq n\cdot 2^{n-1}$. For $n=2$, we establish that for every positive integer $m$, there exists a prime divisor $p$ of $P_{m,2}$ such that ord$_{p} (P_{m,2}) \leq 4$. Consequently, $P_{m,2}$ is never a fifth or higher power. This extends work of Cilleruelo who studied the case $n=1$.
Divisor functions have attracted the attention of number theorists from Dirichlet to the present day. Here we consider associated divisor functions $c_j^{(r)}(n)$ which for non-negative integers $j, r$ count the number of ways of representing $n$ as an ordered product of $j+r$ factors, of which the first $j$ must be non-trivial, and their natural extension to negative integers $r.$ We give recurrence properties and explicit formulae for these novel arithmetic functions. Specifically, the functions $c_j^{(-j)}(n)$ count, up to a sign, the number of ordered factorisations of $n$ into $j$ square-…
All integer solutions $\left(M,a,c\right)$ to the problem of the sums of $M$ consecutive cubed integers $\left(a+i\right)^{3}$ ($a>1$, $0\leq i\leq M-1$) equaling squared integers $c^{2}$ are found by decomposing the product of the difference and sum of the triangular numbers of $\left(a+M-1\right)$ and $\left(a-1\right)$ in the product of their greatest common divisor $g$ and remaining square factors $δ^{2}$ and $σ^{2}$, yielding $c=gδσ$. Further, the condition that $g$ must be integer for several particular and general cases yield generalized Pell equations whose solutions allow to find all …
We investigate various questions concerning the reciprocal sum of divisors, or prime divisors, of the Mersenne numbers $2^n-1$. Conditional on the Elliott-Halberstam Conjecture and the Generalized Riemann Hypothesis, we determine $\max_{n\le x} \sum_{p \mid 2^n-1} 1/p$ to within $o(1)$ and $\max_{n\le x} \sum_{d\mid 2^n-1}1/d$ to within a factor of $1+o(1)$, as $x\to\infty$. This refines, conditionally, earlier estimates of Erdős and Erdős-Kiss-Pomerance. Conditionally (only) on GRH, we also determine $\sum 1/d$ to within a factor of $1+o(1)$ where $d$ runs over all numbers dividing $2^n-1$ fo…
A partition of a positive integer $n$ is a representation of $n$ as a sum of a finite number of positive integers (called parts). A trapezoidal number is a positive integer that has a partition whose parts are a decreasing sequence of consecutive integers, or, more generally, whose parts form a finite arithmetic progression. This paper reviews the relation between trapezoidal numbers, partitions, and the set of divisors of a positive integer. There is also a complete proof of a theorem of Sylvester that produces a stratification of the partitions of an integer into odd parts and partitions int…
In 2016, in the work related to Galois representations, Greenberg conjectured the existence of multi-quadratic $p$-rational number fields of degree $2^{t}$ for any odd prime number $p$ and any integer $t \geq 1$. Using the criteria provided by him to check $p$-rationality for abelian number fields, certain infinite families of quadratic, biquadratic and triquadratic $p$-rational fields have been shown to exist in recent years. In this article, for any integer $k \geq 1$, we build upon the existing work and prove the existence of infinitely many prime numbers $p$ for which the imaginary quadrat…
In 2021, Guyer and Mbirika gave two equivalent formulas that computed the greatest common divisor (GCD) of all sums of $k$ consecutive terms in the generalized Fibonacci sequence $\left(G_n\right)_{n \geq 0}$ given by the recurrence $G_n = G_{n-1} + G_{n-2}$ for all $n \geq 2$ with integral initial conditions $G_0$ and $G_1$. In this current paper, we extend their results to the GCD of all sums of $k$ consecutive squares of these numbers. Denoting these GCD values by the symbol $\mathcal{G}_{G_0, G_1}^2\!(k)$, we prove $\mathcal{G}_{G_0, G_1}^2\!(k) = \gcd\left(G_k G_{k+1} - G_0 G_1,\; G_{k+1}…
This paper is intended as a sequel to a paper arXiv:0803.2636 written by four of the coauthors here. In the paper, they proved a stronger form of the Erdős-Mirksy conjecture which states that there are infinitely many positive integers $x$ such that $d(x)=d(x+1)$ where $d(x)$ denotes the number of divisors of $x$. This conjecture was first proven by Heath-Brown in 1984, but the method did not reveal the nature of the set of values $d(x)$ for such $x$. In particular, one could not conclude that there was any particular value $A$ for which $d(x)=d(x+1)=A$ infinitely often. In the previous paper …
A positive integer $n$ is said to be $k$-layered if its divisors can be partitioned into $k$ sets with equal sum. In this paper, we start the systematic study of these class of numbers. In particular, we state some algorithms to find some even $k$-layered numbers $n$ such that $2^αn$ is a $k$-layered number for every positive integer $α$. We also find the smallest $k$-layered number for $1\leq k\leq 8$. Furthermore, we study when $n!$ is a $3$-layered and when is a $4$-layered number. Moreover, we classify all $4$-layered numbers of the form $n=p^αq^βrt$, where $α$, $1\leq β\leq 3$, $p$, $q$, …
We explore the sums of $k$ consecutive terms in the generalized Fibonacci sequence $\left(G_n\right)_{n \geq 0}$ given by the recurrence $G_n = G_{n-1} + G_{n-2}$ for all $n \geq 2$ with integral initial conditions $G_0$ and $G_1$. In particular, we give precise values for the greatest common divisor (GCD) of all sums of $k$ consecutive terms of $\left(G_n\right)_{n \geq 0}$. When $G_0 = 0$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Fibonacci numbers, and when $G_0 = 2$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Lucas numbers. Denoting the GCD of all sum…
In 1966, Shanks and Schmid investigated the asymptotic behavior of the number of positive integers less than or equal to x which are represented by the quadratic form X^2+nY^2. Based on some numerical computations, they observed that the constant occurring in the main term appears to be the largest for n=2. In this paper, we prove that in fact this constant is unbounded as n runs through positive integers with a fixed number of prime divisors.
We propose an algorithm for finding zero divisors in quaternion algebras over quadratic number fields, or equivalently, solving homogeneous quadratic equations in three variables over $\mathbb{Q}(\sqrt{d})$ where $d$ is a square-free integer. The algorithm is randomized and runs in polynomial time if one is allowed to call oracles for factoring integers.
Inspired by a paper of Erik Westzynthius,we build on work of Harlan Stevens and Hans-Joachim Kanold. Let $k \gt 2$ be the number of distinct prime divisors of a positive integer $n$. In 1977, Stevens used Bonferroni inequalities to get an explicit upper bound on Jacobsthal's function $g(n)$, which is related to the size of largest interval of consecutive integers none of which are coprime to $n$. Letting $u(k)$ be the base $2$ $\log$ of this bound, Stevens showed $u(k)$ is $O((\log k)^2)$, improving upon Kanold's exponent $O(\sqrt{k})$. We use elementary methods similar to those of Stevens to …
A commutative semigroup of abstract factorials is defined in the context of the ring of integers. We study such factorials for their own sake, whether they are or are not connected to sets of integers. Given a subset X of the positive integers we construct a "factorial set" with which one may define a multitude of abstract factorials on X. We study the possible equality of consecutive factorials, a dichotomy involving the limit superior of the ratios of consecutive factorials and we provide many examples outlining the applications of the ensuing theory; examples dealing with prime numbers, Fib…
We show how to carry out a sieve of Eratosthenes up to N in space O(N^{1/3} (log N)^{2/3}) and time O(N log N). These bounds constitute an improvement over the usual versions of the sieve, which take space about O(sqrt{N}) and time at least linear on N. We can also apply our sieve to any subinterval of [1,N] of length Omega(N^{1/3} (log N)^{2/3}) in time essentially linear on the length of the interval. Before, such a thing was possible only for subintervals of [1,N] of length Omega(sqrt{N}). Just as in (Galway, 2000), our approach is related to Diophantine approximation, and also has close ti…
In this article we develop an algorithm which computes a divisor of an integer $N$, which is assumed to be neither prime nor the power of a prime. The algorithm uses discrete time heat diffusion on a finite graph. If $N$ has $m$ distinct prime factors, then the probability that our algorithm runs successfully is at least $p(m) = 1-(m+1)/2^{m}$. We compute the computational complexity of the algorithm in terms of classical, or digital, steps and in terms of diffusion steps, which is a concept that we define here. As we will discuss below, we assert that a diffusion step can and should be consid…
