Related papers: Numbers represented by restricted sums of four squ…

In this paper we give unified formulas for the numbers of representations of positive integers as sums of four generalized $m$-gonal numbers, and as restricted sums of four squares under a linear condition, respectively. These formulas are…

We use a representability theorem of G. L. Watson to examine sums of squares in Quaternion rings with integer coefficients. This allows us to determine a large family of such rings where every element expressible as the sum of squares can…

A well-known theorem of Lagrange asserts that every nonnegative integer $n$ can be written in the form $a^2+b^2+c^2+d^2$, where $a,b,c,d \in \mathbb{Z}$. We characterize the values assumed by $a+b+c+d$ as we range over all such…

Recently the author used certain quaternion orders to demonstrate the universality of some quaternary quadratic forms. Here a further study is done on one of these orders analogous to Hurwitz's proof of the formula for the number of…

We prove that the greatest positive integer that is not expressible as a linear combination with integer coefficients of elements of the set $\{n^2,(n+1)^2,\ldots \}$ is asymptotically $O(n^2)$, verifying thus a conjecture of Dutch and…

In this paper we prove that there exist infinitely many integers which can be expressed as a sum of four cubes of polynomials with integer coefficients. We give several identities that express the integers 1 and 2 as a sum of four cubes of…

Well-known results of Lagrange and Jacobi prove that the every $m \in \mathbb N$ can be expressed as a sum of four integer squares, and the number $r(m)$ of such representations can be given by an explicit formula in $m$. In this paper, we…

In the paper we can prove that every integer can be written as the sum of two integers, one perfect square and one squarefree. We also establish the asympotic formula for the number of representations of an integer in this form. The result…

It is conjectured that every integer N>454 is the sum of seven nonnegative cubes. We prove the conjecture when N is a multiple of 4.

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…

In this paper, it is proved that every sufficiently large even integer can be represented as the sum of two squares of primes, two cubes of primes, two biquadrates of primes and 16 powers of 2. Furthermore, there are at least 5.313% odd…

In this paper we establish some new results similar to Lagrange's four-square theorem. For example, we prove that any integer $n>1$ can be written as $w(5w+1)/2+x(5x+1)/2+y(5y+1)/2+z(5z+1)/2$ with $w,x,y,z\in\mathbb Z$. Let $a$ and $b$ be…

Let $r_Q(n)$ be the representation number of a nonnegative integer $n$ by the quaternary quadratic form $Q=x_1^2+2x_2^2+x_3^2+x_4^2+x_1x_3+x_1x_4+x_2x_4$. We first prove the identity $r_Q(p^2n)=r_Q(p^2)r_Q(n)/r_Q(1)$ for any prime $p$…

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…

We prove part of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n=x^2+Ny^2 for a squarefree integer N.

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 paper, we prove that for $d=3,\dots,8$, every natural number can be written as $t_x+t_y+3t_z+dt_w$, where $x$, $y$, $z$, and $w$ are nonnegative integers and $t_k=k(k+1)/2$ $(k=0,1,2,\ldots)$ is a triangular number. Furthermore, we…

For an integer $x$ let $t_x$ denote the triangular number $x(x+1)/2$. Following a recent work of Z. W. Sun, we show that every natural number can be written in any of the following forms with $x,y,z\in\Z$: $$x^2+3y^2+t_z, x^2+3t_y+t_z,…

Suppose $k$ is a positive integer. In this work, we establish formulas for for the number of representations of integers by the quadratic forms $$ x_{1}^{2}+\cdots+x_{k}^{2}+l\left(x_{k+1}^{2}+\cdots+x_{2k}^{2}\right) $$ for $l\in\{2,4\}$.

We establish that almost every positive integer $n$ is the sum of four cubes, two of which are at most $n^{\theta}$, as long as $\theta\geq192/869$. An asymptotic formula for the number of such representations is established when…