6 problems

Let $\pi$ denote the usual circular constant, and let $x,y,z$ be positive integers. Consider the quantity $|z^\pi-y^\pi-x^\pi|$. Finiteness conjecture for the $\pi$-Fermat expressi…

Let $N\geq 3$, let $(a,b)\in I_N$, where $I_N\subset(\mathbb{Z}/N\mathbb{Z})^{\oplus 2}$ consists of pairs with $a,b,a+b\neq 0$, and set $k:=\varphi(N)/2$. Let $X_N^{[a,b]}$ be the…

Eskandari–Murty's conjecture. The cycle

Fermat's conjecture over an arithmetic function field. For every arithmetic function field $K$, there exists a positive integer $N_0$ depending on $K$ such that $F_N$ has Fermat's…

Let $F_{N,M}$ be the Fermat curve defined by … and let $F^*_{2,4}$ be the curve $z^2=w^4+1$. Let $\sigma=\sigma_1$, so $\sigma(t)=t^p$. Fermat and related curve conjectures. There…

The BSD special-value conjecture. If $L(j_N^{a,b},s)$ does not vanish at $s=1$, then