5 problems

Let $B_{n,k}$ denote the Catalan-triangle quantities used in the paper. For positive integers $r,s,t$ satisfying $r+s+t\equiv 1\pmod 2$, triple-product divisibility conjecture. The…

Let $B_{m,k}$ and $B_{n,k}$ denote the Catalan-triangle quantities used in the paper. Let $m,n,s,t\geqslant 1$ and $r\geqslant 0$, with $r+s+t\equiv 1\pmod 2$. Mixed-product divisi…

Let $B_{n,k}$ denote the Catalan-triangle quantities used in the paper, and let $n,r,s\geqslant 1$. Even-exponent Catalan-triangle congruence conjecture. … This is a complementary…

Let $B_{n,k}$ denote the Catalan-triangle quantities used in the paper, let $r\geqslant 0$ and $s\geqslant 1$, and assume $n\geqslant 4^s-1$. Power-of-four threshold conjecture. ……

Let $B_{n,k}$ denote the Catalan-triangle quantities used in the paper, and let $n,r\geqslant 1$. Odd-power Catalan-triangle congruence conjecture. … if and only if $n=2^a-2^b$ for…