Related papers: Sums of triangular numbers and sums of squares

Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb N$ let $N(a,b,c,d;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2+dw^2$, and let $t(a,b,c,d;n)$ be the…

Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb N$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2$…

For positive integers $a,b,c$, and an integer $n$, the number of integer solutions $(x,y,z) \in \mathbb Z^3$ of $a \frac{x(x-1)}{2} + b \frac{y(y-1)}{2} + c \frac{z(z-1)}{2} = n$ is denoted by $t(a,b,c;n)$. In this article, we prove some…

We investigate here the representability of integers as sums of triangular numbers, where the $n$-th triangular number is given by $T_n = n(n + 1)/2$. In particular, we show that $f(x_1,x_2,..., x_k) = b_1 T_{x_1} +...+ b_k T_{x_k}$, for…

Recently, Jha (arXiv:2007.04243, arXiv:2011.11038) has found identities that connect certain sums over the divisors of $n$ to the number of representations of $n$ as a sum of squares and triangular numbers. In this note, we state a…

In this paper, we study universal sums of triangular numbers and squares. Specifically, we prove that a sum of triangular numbers and squares is universal if and only if it represents…

We give formulas for the number of representations of non negative integers by various quadratic forms. We also give evaluations in the case of sum of two cubes (cubic case) and the quintic case, as well. We introduce a class of generalized…

We examine what integers are representable as sums of three cubes. We also provide formulas for the number of representations of $x^3+y^3+z^3=n$ under the condition $x+y+z=t$. Also we show how the problem of three cubes is related to…

Let $\sum_{d|n}$ denote sum over divisors of a positive integer $n$, and $t_{r}(n)$ denote the number of representations of $n$ as a sum of $r$ triangular numbers. Then we prove that $$…

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.

Let $r_{k}(n)$ denote the number of representations of the integer $n$ as a sum of $k$ squares. In this paper, we give an asymptotic for $r_{k}(n)$ when $n$ grows linearly with $k$. As a special case, we find that \[ r_{n}(n) \sim \frac{B…

A number of the form $x(x+1)/2$ where $x$ is an integer is called a triangular number. Suppose, $N(a_1,\cdots,a_k;n)$ and $T(a_1,\cdots,a_k;n)$ denote the number of ways $n$ can be expressed as $\sum_{i=1}^k a_ix_i^2$ and $\sum_{i=1}^k…

Let $B$ be the set of odd integers that are sums of two coprime squares. We prove that the trigonometric polynomial $S(\alpha;N)=\sum_{b\in B,b\leq N} e(b\alpha)$ satisfies \[ \frac{S(\alpha; N)}{N/\sqrt{\log N}}<<_{A,A'} \frac{1}{\phi(q)}…

Let $r_3(n)$ be the number of representations of a positive integer $n$ as a sum of three squares of integers. We give two distinct proofs of a conjecture of Wagon concerning the asymptotic value of the mean square of $r_3(n)$.

We prove that a positive integer not of the form, 4^{k}(8m+7) can be expressible as a sum of three or fewer squares by using some results of Kane and Sun on mixed sums of squares and triangular numbers.

Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a_1,a_2,\ldots,a_k,n\in\Bbb N$ let $N(a_1,a_2,\ldots,a_k;n)$ be the number of representations of $n$ by…

In this paper, we derive a general formula to express the product of three theta functions as a linear combination of other products of three theta functions. Moreover, we use the main formula to deduce a general formula for the product of…

Let $a_k(n)$ denotes the number of representations of a non-negative integer $n$ as sum of $k$ quadratic forms of the type $x^2+xy+y^2$ and $a_{\lambda_1,\lambda_2,\lambda_3\dots\lambda_k}(n)$ denotes the number of representations $n$ as a…

I discuss a variety of results involving s(n), the number of representations of n as a sum of three squares. One of my objectives is to reveal numerous interesting connections between the properties of this function and certain modular…

Which integers can be written as a quotient of sums of distinct powers of three? We outline our first steps toward an answer to this question, beginning with a necessary and almost sufficient condition. Then we discuss an algorithm that…