4 problems

Infinitude conjecture. There exist infinitely many $(n_1,n_2,n_3)\in\mathbb N^3$ with $n_1,n_2,n_3\geq2$ such that all four displayed integers are perfect squares.

Let $T(\alpha)$ be the set defined in the paper, and write $T_{\leq x}(\alpha)$ for its elements up to $x$. Logarithmic growth conjecture. For suitable $\alpha\in(0,1)$, there exis…

Henk–Weismantel conjecture. Every minimal solution lies in this convex hull:

He–Shiue–Venkat conjecture. For any solution $(\hat{x},\hat{y},\hat{z})$ of $ax+by+cz=n$, there exist $s_i\in\hat{S_i}$ such that