5 problems

Finite test set conjecture. A sum of triangular numbers $f$ represents all odd integers if and only if it represents the integers

For positive integers $a_1,\a_2,\a_3$, write $\Delta_{a_1,a_2,a_3}$ for the corresponding sum of triangular numbers; in particular, $\Delta_{1,4,5}$ is the ternary sum with coeffic…

Very triangular arithmetic-progression conjecture. For any positive integer $k$, there exists an arithmetic progression $AP(k)$ of length $k$ such that $t_j\in VT$ for all…

Binary-weight conjecture. If $n\in\mathbb{N}$ is such that $|(n)_2^1|\geq 6$, then $|(t_n)_2^1|\geq 4$.

Let $\mathbb{N}_0$ denote the set of nonnegative integers. Consider the two forms … and … where $a,b,c\in\mathbb{N}_0$. Three-form representation conjecture. Every natural number o…