Related papers: On practical numbers of some special forms

Generalised Pythagorean triples are integer tuples $(x,y,z)$ satisfying the equation $E_{a,b,c}: ax^2+by^2+cz^2=0$. A significant amount of research has been devoted towards understanding generalised Pythagorean triples and, in particular,…

We show that if $A$ and $B$ are finite sets of real numbers, then the number of triples $(a,b,c)\in A\times B\times (A\cup B)$ with $a+b=2c$ is at most $(0.15+o(1))(|A|+|B|)^2$ as $|A|+|B|\to\infty$. As a corollary, if $A$ is antisymmetric…

We study whether sufficiently large integers can be written in the form cp+T_x, where p is either zero or a prime congruent to r mod d, and T_x=x(x+1)/2 is a triangular number. We also investigate whether there are infinitely many positive…

In this paper, we extend recent work of the third author and Ziegler on triples of integers $(a,b,c)$, with the property that each of $(a,b,c)$, $(a+1,b+1,c+1)$ and $(a+2,b+2,c+2)$ is multiplicatively dependent, completely classifying such…

Problem 2 at the 56th International Mathematical Olympiad (2015) asks for all triples (a,b,c) of positive integers for which ab-c, bc-a, and ca-b are all powers of 2. We show that this problem requires only a primitive form of arithmetic,…

In this paper, we study the number of representations of a positive integer $n$ by two positive integers whose product is a multiple of a polygonal number.

In this paper we first investigate for what positive integers $a,b,c$ every nonnegative integer $n$ can be represented as $x(ax+1)+y(by+1)+z(cz+1)$ with $x,y,z$ integers. We show that $(a,b,c)$ can be either of the following seven triples:…

In this paper, we proved that there are infinite cube--free numbers of the form $[n^c]$ for any fixed real number $1<c<11/6$.

The abc conjecture, one of the most famous open problems in number theory, claims that three positive integers satisfying a+b=c cannot simultaneously have significant repetition among their prime factors; in particular, the product of the…

In this paper we mainly study sums of four rational squares with certain restrictions. Let $\mathbb Q_{\ge0}$ be the set of nonnegative rational numbers. We establish the following four-square theorem for rational numbers: For any…

Let $a,b,c$ be positive integers. It is known that there are infinitely many positive integers not representated by $ax^2+by^2+cz^2$ with $x,y,z\in\mathbb Z$. In contrast, we conjecture that any natural number is represented by $\lfloor…

We give a~detailed construction of the complete ordered field of real numbers by means of infinite decimal expansions. We prove that in the canonical encoding of decimals neither addition nor multiplication is {\em computable}, but that…

We consider the triples of integer numbers that are solutions of the equation $x^2+qy^2=z^2$, where $q$ is a fixed, square-free arbitrary positive integer. The set of equivalence classes of these triples forms an abelian group under the…

Following Srinivasan, an integer n\geq 1 is called practical if every natural number in [1,n] can be written as a sum of distinct divisors of n. This motivates us to define f(n) as the largest integer with the property that all of 1, 2,…

We explore primitive Pythagorean triples of special forms $(a,b,b+g)$ and $(a,a+f,c)$, with $g,f\in\mathbb{Z}^+$. For each $g$ and $f$, we provide a method to generate infinitely many such primitive triples. Lastly, for each $g$, we…

In this paper, we determine all the positive integers $a, b$ and $c$ such that every nonnegative integer can be represented as $$ f^{a,b}_c(x,y,z,w)=ax^2+by^2+c(z^2+zw+w^2) \,\, \textrm{with} \,\,x,y,z,w\in\mathbb{Z}. $$ Furthermore, we…

In this paper we use the theory of modular forms to find formulas for the number of representations of a positive integer by certain class of quadratic forms in eight variables, viz., forms of the form $a_1x_1^2 + a_2 x_2^2 + a_3 x_3^2 +…

For an integer $b\geq 2$, a positive integer is called a $b$-Niven number if it is a multiple of the sum of the digits in its base-$b$ representation. In this article, we show that every arithmetic progression contains infinitely many…

Let $a$, $b$, $c$ be fixed coprime positive integers with $\min\{ a,b,c \} >1$. Let $N(a,b,c)$ denote the number of positive integer solutions $(x,y,z)$ of the equation $a^x + b^y = c^z$. We show that if $(a,b,c)$ is a triple of distinct…

For $m=3,4,\ldots$, the polygonal numbers of order $m$ are given by $p_m(n)=(m-2)\binom n2+n\ (n=0,1,2,\ldots)$. For positive integers $a,b,c$ and $i,j,k\ge3$ with $\max\{i,j,k\}\ge5$, we call the triple $(ap_i,bp_j,cp_k)$ universal if for…