5 problems

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…

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…