20 problems

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 $p$ 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…

Divisibility conjecture. For every such $x$, $y$, and $p$, the integer $H_p(x,y)$ is not divisible by $p^2$.

Let $p>5$ be prime and let $t\equiv -x/y\pmod p$ arise from a putative first-case solution $x^p+y^p+z^p=0$ with $x,y,z\in\mathbb{Z}-\{0\}$ mutually coprime and…

Let $p>5$ be prime, let $x,y,z\in\mathbb{Z}-\{0\}$ be mutually coprime and satisfy $x^p+y^p+z^p=0$ with $xyz\not\equiv 0\pmod p$, and let $t\equiv -x/y\pmod p$. Let the polynomial…

Let $p>5$ be prime, let $x,y,z\in\mathbb{Z}-\{0\}$ be mutually coprime and satisfy $x^p+y^p+z^p=0$ with $xyz\not\equiv 0\pmod p$, and let $t\equiv -x/y\pmod p$. Consider the explic…

Let $E$ denote the Frey elliptic curve associated with a putative nontrivial solution of Fermat's equation. Serre's epsilon-conjecture. The conjecture that $E$ is not modular was n…

Frey's conjecture. This elliptic curve would not be modular; equivalently, it would be a counterexample to the Shimura–Taniyama–Weil conjecture.

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…

Fermat's Last Theorem is an arithmetical statement concerning solutions of the equation $x^n+y^n=z^n$ in positive integers for exponents $n>2$. Friedman's conjecture. Fermat's Last…

Let $p>3$ be prime. For a prime $q\equiv3\pmod4$, write $f$ for the order of $q$ modulo $p$ and set $\kappa=(q^f-1)/p$. Let $\xi=e^{2\pi i/n}$ for each integer $n>2$ dividing $q-1$…

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. For a $p$-principal prime $q\equiv3\pmod 4$, let $f$ be the order of $q$ modulo $p$ and set $\kappa=(q^f-1)/p$. Furtwängler-type Kummer criterion. There exists…

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 $q$…

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

Prime-polynomial conjecture. In case (i), $u+v\not\equiv0\pmod p$, infinitely many primes $q$ with $\kappa\not\equiv0\pmod p$ make $A_p(\rho)X^p+\cdots+A_0(\rho)$ reducible modulo…

Inertia conjecture. The set of primes $q\in Q_\rho$ having $\rho$-inertia for $\widehat{\mathcal F}_n$ is infinite.

Divisor-function conjecture. There exists such a divisor function $m$ for which infinitely many primes $q$, satisfying $\kappa\not\equiv0\pmod p$ and totally split in $F_{m(q-1)}$,…

Infinitude conjecture. The number of such primes $\ell$ tends to infinity with $p$. This would provide many ineffective Wieferich criteria. The claim is presented as an experimenta…

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