Related papers: Mixed sums of triangular numbers and certain binar…

We study three classes of combinatorial sums involving central binomial coefficients and harmonic numbers, odd harmonic numbers, and even indexed harmonic numbers, respectively. In each case we use summation by parts to derive recursive…

We introduce a full binary directed tree structure to represent the set of natural numbers, further categorizing them into three distinct subsets: pure odd numbers, pure even numbers, and mixed numbers. We adopt a binary string…

Let $ \prod_{i=1}^d (X-\alpha_i Y) \in{\mathbb C}[X,Y]$ be a binary form and let $\epsilon_1,\dots,\epsilon_d$ be nonzero complex numbers. We consider the family of binary forms $ \prod_{i=1}^d (X-\alpha_i \epsilon_i^aY)$, $a\in {\mathbb…

Using modular forms we determine formulas for the number of representations of a positive integer by diagonal octonary quadratic forms with coefficients $1$, $2$, $3$ or $6$.

In this note we show that the system of equations t_{x}+t_{y}=t_{p},\quad t_{y}+t_{z}=t_{q},\quad t_{x}+t_{z}=t_{r}, where $t_{x}=x(x+1)/2$ is a triangular number, has infinitely many solutions in integers. Moreover we show that this system…

For a natural number $m$, generalized $m$-gonal numbers are those numbers of the form $p_m(x)=\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in \mathbb Z$. In this paper we establish conditions on $m$ for which the ternary sum $p_m(x)+p_m(y)+p_m(z)$ is…

We study triples {a,b,c} of distinct nonzero rational numbers such that a+1,b+1,c+1,ab+1,ac+1,bc+1 and abc+1 are all perfect squares. We prove that there exist infinitely many such triples. In contrast, we show that no triple of positive…

We introduce a full solution to a problem considered by Wang and Chu concerning series involving the squares of finite sums of the form $1 + \frac{1}{3}+ \cdots + \frac{1}{2n-1}$. Our proof involves techniques from the theory of colored…

We show that for a large class of cubic polynomials $f$, every sufficiently large number can be written as a sum of seven positive values of $f$. As a special case, we show that every number greater than $e^{10^7}$ is a sum of seven…

In this paper, we study partitions of positive integers with restrictions involving squares. We mainly establish the following two results (which were conjectured by Sun in 2013): (i) Each positive integer $n$ can be written as $n=x+y+z$…

We introduce and study arithmetic polygons. We show that these arithmetic polygons are connected to triples of square pyramidal numbers. For every odd $N\geq3$, we prove that there is at least one arithmetic polygon with $N$ sides. We also…

In this article, we make use of some known method to investigate some properties of the numbers represented as sums of two equal odd powers, i.e., the equation $x^n+y^n=N$ for $n\ge3$. It was originated in developing algorithms to search…

In this paper, we consider mixed sums of generalized polygonal numbers. Specifically, we obtain a finiteness condition for universality of such sums; this means that it suffices to check representability of a finite subset of the positive…

In the paper, some special linear combinations of the terms of rational cycles of generalized Collatz sequences are studied. It is proved that if the coefficients of the linear combinations satisfy some conditions then these linear…

I want to show one possibility to proof the Collatz conjecture, also called 3n+1 conjecture, for any natural number N. For this, I limit my analysis on the direct odd follower of every natural odd number and show the connections between the…

The binomial coefficients and Catalan triangle numbers appear as weight multiplicities of the finite-dimensional simple Lie algebras and affine Kac--Moody algebras. We prove that any binomial coefficient can be written as weighted sums…

We show that many of Ramanujan's modular equations of degree 3 can be interpreted in terms of integral ternary quadratic forms. This way we establish that for any n in N |{n= x(x+1)/2 + y^2 +z^2 : x,y,z in Z}| >= |{n= x(x+1)/2 + 3y^2 +3z^2:…

Permutations can be represented as linear combinations of natural numbers with different powers. In this paper, its coefficient matrix and inverse matrix is derived, and the results show the coefficient matrix is a lower triangular matrix…

A set of $m$ positive integers $\{x_{1},\ldots,x_{m}\}$ is called a $P^{3}_{1}$-set of size $m$ if the product of any three elements in the set increased by one is a cube integer. A $P^{3}_{1}$-set $S$ is said to be extendible if there…

The problem of finding the sum of a polynomial's values is considered. In particular, for any $n\geq 3$, the explicit formula for the sum of the $n$th powers of natural numbers $S_n=\sum_{x=1}^{m}x^{n}$ is proved:…