9 problems

Let $\bold Q$ denote the field of rational numbers, and call an elliptic curve over $\bold Q$ modular if it admits a holomorphic map from some modular curve $X_0(N)$ onto it. Taniy…

Irrational-exponent conjecture. Then $n$ is irrational. Equivalently, no such equation has a rational exponent $n$ under these hypotheses. This statement is presented after the pap…

Let $p$ be an odd prime, let $K=\mathbb Q(\zeta)$ be the $p$-th cyclotomic field, and consider a solution of the SFLT2 equation. Reduced-form conjecture. If the SFLT2 conjecture fa…

Let $p>3$ be prime. For a triple $(p,u,v)\in\mathbb Z^3$ with $u$ and $v$ coprime and $p\mid v$, let $q$ be a $p$-principal prime with $puv\not\equiv0\pmod q$, let $n$ be the order…

Let $p>3$ be prime, let $K=\mathbb Q(\zeta)$ be the $p$-th cyclotomic field, and let $u,v\in\mathbb Z\setminus\{0\}$ be coprime. The equation … according as $u+v\not\equiv0\pmod p$…

Let $p>3$ be prime and let $(p,u,v)\in\mathbb Z^3$ with $u$ and $v$ coprime and $p\mid v$. For a $p$-principal prime $q$ with $q\nmid puv$, let $n$ be the order of $v/u$ modulo $q$…

Let $p$ be an odd prime, set $K=$\mathbb Q$(\zeta)$ and $\mathfrak p=(\zeta-1)$\mathbb Z$[\zeta]$. For coprime integers $u,v$, let $\delta\geq 0$ and let $\mathfrak w_1$…

Order conjecture. There exists such a solution $(u,v)$ for which the order of $v/u$ modulo $q$ is at least $m$.

The cyclotomic ideal-power conjecture. This equation has no solution except in the trivial cases