Related papers: Representations of integers as sums of four polygo…

The paper presents a discussion on the asymptotic formula for the number of plane partitions of a large positive integer.

Practical numbers are positive integers $n$ such that every positive integer less than or equal to $n$ can be written as a sum of distinct positive divisors of $n$. In this paper, we show that all positive integers can be written as a sum…

In this paper, we classify the $(a_1,a_2,a_3,a_4,a_5)$ for which the universality of an $m$-gonal form $F_m(\mathbf x)$ having its first five coefficients as $(a_1,a_2,a_3,a_4,a_5)$ is characterized as the representability of positive…

We identify a partition-theoretic generalization of Riemann zeta function and the equally positive integer-indexed harmonic sums at infinity, to obtain the generating function and the integral representations of the latter. The special…

Every natural number greater than two may be written as the sum of a prime and a square-free number. We establish several generalisations of this, by placing divisibility conditions on the square-free number.

We study an asymptotic formula for counting integral points over an equation defined by a non-degenerated indefinite integral ternary quadratic form $f$ representing a non-zero integer $a$ such that $-a\cdot det(f)$ is square over a number…

In this paper, we prove some results of restricted sums of four squares using arithmetic of quaternions in the ring of Lipschitz integers. For example, we show that every nonnegative integer $n$ can be written as $x^{2}+y^{2}+z^{2}+t^{2}$…

We evaluate the convolution sums $\sum_{l,m\in {\mathbb N}, {l+15m=n}} \sigma(l) \sigma(m)$ and $\sum_{l,m\in {\mathbb N}, {3l+5m=n}} \sigma(l) \sigma(m)$ for all $n\in {\mathbb N}$ using the theory of quasimodular forms and use these…

In this paper, we introduce a natural geometric extension of the partition function. More precisely, we investigate the problem of counting partitions of a rectangle into rectangular blocks with integer sides. Here, two partitions of a…

This work presents a formalization of analogy on numbers that relies on generalized means. It is motivated by recent advances in artificial intelligence and applications of machine learning, where the notion of analogy is used to infer…

Representations of nonnegative polynomials as sums of squares are central to real algebraic geometry and the subject of active research. The sum-of-squares representations of a given polynomial are parametrized by the convex body of…

Consider an irreducible finite Coxeter system. We show that for any nonnegative integer n the sum of the nth powers of the Coxeter exponents can be written uniformly as a polynomial in four parameters: h (the Coxeter number), r (the rank),…

We give a new proof of Milne's formulas for the number of representations of an integer as a sum of 4m^2 and 4m(m+1) squares. The proof is based on explicit evaluation of pfaffians with elliptic function entries, and relates Milne's…

In this article, we consider the problem of determining formulas for the number of representations of a natural number $n$ by a sum of figurate numbers with certain positive integer coefficients. To achieve this, we prove that the…

Sylvester showed that the partition of an integer into a set of positive integers can be represented as a sum of the polynomial term and quasiperiodic components called the Sylvester waves. The wave itself is a weighted sum of the…

Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition $S+T=\makeset{m+n}{m \in S, \: n \in T}$ and…

The general formulas for finding the quantity of all primitive and nonprimitive triples generated by the given number x have been proposed. Also the formulas for finding the complete quantity of the representations of the integers as a…

In this paper we shall investigate the problem of the representation of the number of integral points of an elliptic curve modulo a prime number p. We present a way of expressing an exponential sum which involves polynomials of third…

We investigate the representation of symmetric polynomials as a sum of squares. Since this task is solved using semidefinite programming tools we explore the geometric, algebraic, and computational implications of the presence of discrete…

We study representations of integers as sums of the form $\pm a_1\pm a_2\pm \dotsb \pm a_n$, where $a_1,a_2,\ldots$ is a prescribed sequence of integers. Such a sequence is called an Erd\H{o}s-Sur\'anyi sequence if every integer can be…