8 problems

Hasler's structural infinitude conjecture. The equation $\binom{3n}{n} \equiv 3^n \pmod{n}$ has infinitely many solutions of the form $n=3^k\cdot p$, where $k$ is a positive intege…

Generalized divisibility conjecture. is congruent to zero modulo

Let $A_{n,k}$ be the ballot numbers … For all $n,r_1,\ldots,r_m\in\mathbb{Z}^+$ such that $r_1+\cdots+r_m\equiv1\pmod2$, and with $\varepsilon=\pm1$, Guo–Zeng's general ballot-prod…

Let $A_{n,k}$ be the ballot numbers … For all $n,r,s,t\in\mathbb{Z}^+$ with $r+s+t\equiv1\pmod2$, and with $\varepsilon=\pm1$, Guo–Zeng's three-factor ballot-number conjecture. The…

Let $A_{n,k}$ be the ballot numbers … For $r\in\mathbb{N}$, $s,t\in\mathbb{Z}^+$ with $r+s+t\equiv1\pmod2$, $n\in\mathbb{Z}^+$, and $\varepsilon=\pm1$, Guo–Zeng's adjacent-index co…

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…