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A positive quadratic form is $(k,\ell)$-universal if it represents all the numbers $kx+\ell$ where $x$ is a non-negative integer, and almost $(k,\ell)$-universal if it represents all but finitely many of them. We prove that for any $k,\ell$…

Let $a,b,c,d,e,f\in\mathbb N$ with $a\ge c\ge e>0$, $b\le a$ and $b\equiv a\pmod2$, $d\le c$ and $d\equiv c\pmod2$, $f\le e$ and $f\equiv e\pmod2$. If any nonnegative integer can be written as $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$ with…

An integer of the form $p_m(x)= \frac{(m-2)x^2-(m-4)x}{2} \ (m\ge 3)$, for some integer $x$ is called a generalized polygonal number of order $m$. A ternary sum $\Phi_{i,j,k}^{a,b,c}(x,y,z)=ap_{i+2}(x)+bp_{j+2}(y)+cp_{k+2}(z)$ of…

In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than 2719 can be represented by the famous Ramanujan form $x^2+y^2+10z^2$, equivalently the…

For each integer $m\ge3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. Given positive integers $a,b,c,k$ and an odd prime number $p$ with $p\nmid c$, we employ the theory of ternary…

For each integer $x$, the $x$-th generalized pentagonal number is denoted by $P_5(x)=(3x^2-x)/2$. Given odd positive integers $a,b,c$ and non-negative integers $r,s$, we employ the theory of ternary quadratic forms to determine when the sum…

For any $m\ge3$, every integer of the form $p_m(x)=\frac{(m-2)x^2-(m-4)x}2$ with $x \in \z$ is said to be a generalized $m$-gonal number. Let $a\le b\le c$ be positive integers. For every non negative integer $n$, if there are integers…

An integer-valued polynomial $P(x,y,z)$ is said to be universal (over $\mathbb Z$) if each nonnegative integer can be written as $P(x,y,z)$ with $x,y,z\in\mathbb Z$. In this paper, we mainly introduce a new technique to determine the…

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…

In 2016, while studying restricted sums of integral squares, Sun posed the following conjecture: Every positive integer $n$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb{N}=\{0,1,\cdots\})$ with $x+3y$ a square. Meanwhile, he also…

An integer of the form $P_8(x)=3x^2-2x$ for some integer $x$ is called a generalized octagonal number. A quaternary sum $\Phi_{a,b,c,d}(x,y,z,t)=aP_8(x)+bP_8(y)+cP_8(z)+dP_8(t)$ of generalized octagonal numbers is called {\it universal} if…

A (positive definite and integral) quadratic form is said to be $\textit{prime-universal}$ if it represents all primes. Recently, Doyle and Williams in [2] classified all prime-universal diagonal ternary quadratic forms, and all…

For any integer $x$, let $T_x$ denote the triangular number $\frac{x(x+1)}{2}$. In this paper we give a complete characterization of all the triples of positive integers $(\alpha, \beta, \gamma)$ for which the ternary sums $\alpha x^2…

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…

The 1-3-5 conjecture of Z.-W. Sun states that any $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as $x^2+y^2+z^2+w^2$ with $w,x,y,z\in\mathbb N$ such that $x+3y+5z$ is a square. In this paper, via the theory of ternary quadratic forms and…

Quadratic forms over Z that represent all positive integers are called universal. Starting with Ramanujan, 54 universal quaternary quadratic forms without cross product terms were discovered. The form that is the sum of four squares was…

In this paper, by using the arithmetic theory of ternary quadratic forms, we study some refinements on Lagrange's four-square theorem. For example, given positive integers $a,b$ satisfying some algebraic conditions and a positive integer…

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 $m=3,4,\ldots$, 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…

A quadratic polynomial $\Phi_{a,b,c}(x,y,z)=x(ax+1)+y(by+1)+z(cz+1)$ is called universal if the diophantine equation $\Phi_{a,b,c}(x,y,z)=n$ has an integer solution $x,y,z$ for any non negative integer $n$. In this article, we show that if…