Related papers: On Waring's Problem: Two Cubes and Two Minicubes

We consider several old problems involving the number of prime divisors function $\omega(n)$, as well as the related functions $\Omega(n)$ and $\tau(n)$. Firstly, we show that there are infinitely many positive integers $n$ such that…

We study the representations of large integers $n$ as sums $p_1^2 + ... + p_s^2$, where $p_1,..., p_s$ are primes with $| p_i - (n/s)^{1/2} | \le n^{\theta/2}$, for some fixed $\theta < 1$. When $s = 5$ we use a sieve method to show that…

Let $k \geq 2$ and $b \geq 3$ be integers, and suppose that $d_1, d_2 \in \{0,1,\dots, b - 1\}$ are distinct and coprime. Let $\mathcal{S}$ be the set of non-negative integers, all of whose digits in base $b$ are either $d_1$ or $d_2$. Then…

In this paper I demonstrate that any pair (m, n) of non-zero and distinct rational numbers may have, at most, four representations as the product of two rational factors such that the sum of factors of m coincides with the sum of factors of…

We have proved in this paper that numbers can be expressed in algebraic form using one variable and two real rational quantities and thus sum of three cubes can also be expressed in algebraic form as a cubic polynomial. Using skeletal or…

An arithmetic formula is an expression involving only the constant $1$, and the binary operations of addition and multiplication, with multiplication by $1$ not allowed. We obtain an asymptotic formula for the number of arithmetic formulas…

Following Bhargava and Hanke's celebrated 290-theorem, we prove a universality theorem for all positive-definite integer-valued quadratic forms that represent all positive integers coprime to $3$. In particular, if a positive-definite…

This paper is concerned with finite sequences of integers that may be written as sums of squares of two nonzero integers. We first find infinitely many integers $n$ such that $n, n+h$ and $n+k$ are all sums of two squares where $h$ and $k$…

A double-base representation of an integer n is an expression n = n_1 + ... + n_r, where the n_i are (positive or negative) integers that are divisible by no primes other than 2 or 3; the length of the representation is the number r of…

We investigate the problem of the maximum number of cubic subwords (of the form $www$) in a given word. We also consider square subwords (of the form $ww$). The problem of the maximum number of squares in a word is not well understood.…

For a set $A$ of nonnegative integers, let $R_2(A,n)$ and $R_3(A,n)$ denote the number of solutions to $n=a+a'$ with $a,a'\in A$, $a<a'$ and $a\leq a'$, respectively. In this paper, we prove that, if $A\subseteq \mathbb{N}$ and $N$ is a…

Let $n$ be a cubefree natural number and $p\geq 5$ be a prime number. Assume that $n$ is not expressible as a sum of the form $x^3+y^3$, where $x,y\in \mathbb{Q}$. In this note, we study the solutions (or lack thereof) to the equation…

Let $m$ be a positive integer and $b_{m}(n)$ be the number of partitions of $n$ with parts being powers of 2, where each part can take $m$ colors. We show that if $m=2^{k}-1$, then there exists the natural density of integers $n$ such that…

This work proposes a proof of the simplest cubic primes counting problem. It shows that the subset of primes {p = n^3 + 2 is prime : n => 1} is an infinite subset of primes. Further, the expected order of magnitude of the cubic primes…

In this article we further develop methods for representing integers as a sum of three cubes. In particular, a barrier to solving the case $k=3$, which was outlined in a previous paper of the second author, is overcome. A very recent…

An expression is any mathematical formula that contains certain formal variables and operations to be executed in a specified order. In computer science, it is usually convenient to represent each expression in the form of an expression…

We study the asymptotics of the average number of squares (or quadratic residues) in Z_n and Z_n^*. Similar analyses are performed for cubes, square roots of 0 and 1, and cube roots of 0 and 1.

Let $f(n)=\min_{p} |n-p|$, where $p$ is a prime. We show that there is a positive constant $\delta$ such that for any large integer $N$ there exist two positive integers $n_1$ and $n_2$ such that $N=n_1 + n_2$ and $f(n_i)\gg \ln N (\ln\ln…

Let $b(n)$ denote the number of cubic partition pairs of $n$. We give affirmative answer to a conjecture of Lin, namely, we prove that $$b(49n+37)\equiv 0 \pmod{49}.$$ We also prove two congruences modulo $256$ satisfied by…

Let $k\ge 1$ be an integer. We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{k}+p_{2}^{2}+p_{3}^{2}$, where $p_1,p_2,p_3$ are prime numbers, holds in intervals shorter than the ones…