35 problems

Sun's conjecture. The polynomials $x^2+15y^2+P_5(z)$, $x^2+11y^2+P_5(z)$, $x^2+7y^2+2P_5(z)$, and $4x^2+3y^2+P_5(z)$ should represent every natural number over $\mathbb Z$.

Sun's conjecture. Every natural number other than $216$ can be written as

Let $p_8$ denote the generalized octagonal-number polynomial, and let $b,c,d$ be chosen from the five triples … A sum $p_8+bp_8+cp_8+dp_8$ is called universal over $\mathbb{Z}$ whe…

Let $P_8(x)=3x^2-2x$ for an integer $x$, and for positive integers $a\leq b\leq c\leq d$ define … A sum is universal over $\mathbb Z$ if the equation … has an integer solution for…

Sun's universality conjecture. Every nonnegative integer $n$ can be written as

Let $p_7$ denote the generalized heptagonal-number function and let $\mathbb N$ denote the nonnegative integers. Weighted heptagonal exceptional-set conjecture. The following two i…

Signed pentagonal universality conjecture. For each $r\in\{\pm1,\pm3\}$, every integer $n>N(r)$ can be written as

Let $\mathbb N$ denote the nonnegative integers. Four-term pentagonal representation conjecture. Every integer $n>51$ can be written as … with $w,x,y,z\in\mathbb N$. This is an exp…

Let $p_m(x)$ and $\bar p_m(x)$ denote the generalized polygonal-number functions used in the paper, and let $\mathbb N$ denote the nonnegative integers. Eventual representation con…

Three-term nonnegative representation conjecture. Every $n\in\mathbb N$ can be written as

Conjecture on two polygonal summands. For $s\in\{4,5,6,7,8,10\}$, all natural numbers can be written as a sum of a practical number and two $s$-gonal numbers.

Conjecture on practical and polygonal sums. If $s>3$, $s \not\equiv 0\pmod{12}$, and $s \not\equiv 4\pmod{12}$, then all sufficiently large natural numbers can be written as a sum…

Sun's conjecture. For every $n\in\mathbb{N}$, the equation

Let $P(\ell,r)$ denote the $\ell$-gonal numbers, and consider solutions to … where $P$ and $P'$ are $\ell$-gonal numbers. In the example under discussion, $\ell=6$ and…

Let $P(\ell,r)$ denote the $\ell$-gonal numbers, and consider solutions to … where $P$ and $P'$ are $\ell$-gonal numbers. In the example under discussion, $\ell=5$ and…

Let $P(\ell,r)=\frac{r((\ell-2)r-(\ell-4))}{2}$ denote the $\ell$-gonal numbers, and consider the problem of finding $\ell$-gonal numbers $P$ and $P'$ satisfying $P=mP'$. Suppose t…

Sun's conjecture. For any integer $m\ge 3$, the sum

Hirose's conjecture. The number

Let $n$ be an integer greater than $1$. Write a triangular number as $T_r=r(r+1)/2$. Two-triangular-two-fifth-powers conjecture. Every $n$ can be written … for nonnegative integers…

For $k\geq 3$, define the generalized $k$-gonal numbers by … In particular, $p_8(x)$ is a generalized octagonal number. Sun's conjecture. … This predicts the exact exceptional set…

Let $p_6(n)=n(2n-1)$ be a hexagonal number, for $n\in\mathbb N=\{0,1,2,\ldots\}$. Sun's conjecture. Every $n\in\mathbb N$ can be written as … for some $x_1,x_2,x_3,x_4\in\mathbb N$…

Let $p_3(n)=n(n+1)/2$ be a triangular number and let $p_6(n)=n(2n-1)$ be a hexagonal number, for $n\in\mathbb N=\{0,1,2,\ldots\}$. Sun's conjecture. Every $n\in\mathbb N$ can be wr…

Let $a,b,c$ be positive integers with $a\le b\le c$. Combined-floor conjecture. If $c>1$, then … If $(a,b,c)\ne(1,1,1),(1,1,3),(1,1,7),(1,3,3)$, then … The conjecture extends the p…

Let $p_m(n)=((m-2)n^2-(m-4)n)/2$ denote the generalized $m$-gonal number. Floored polygonal-sum conjecture. For all positive integers $a,b,c$, every natural number belongs to … The…

Sun's conjecture for $T$. Every integer $n\ge2$ can be expressed as $r+s+t$, where $r,s,t\in T$, $r\le s\le t$, and $s$ is odd. In addition, for each ordered pair $(b,c)$ among…