10 problems

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…

Low-degree avoiding polynomial conjecture. There is such a polynomial $g_0$ with

Let $p\ge5$ be prime, and write … where $m\in\mathbb{Z}_{\ge0}$ and $\delta,\varepsilon\in\{0,1\}$. Let $A_{u,a}^{(r)}(X)$ be the generalized Atkin polynomial associated with the n…

Additive Wieferich-Wilson prime conjecture. If $p>2$ and the greatest common divisor of $[d]$ and

Prime-modulus root equidistribution conjecture. The resulting sequence of normalized roots is uniformly distributed in $[0,1]$.

Let $f\in\mathbf{Z}[X]$ be a monic irreducible polynomial of degree at least $2$. For each prime $p$, let $\varrho_f(p)$ be the number of roots of $f$ modulo $p$, and let…

Let $f\be\in\mathbf{Z}[X]$ be a monic irreducible polynomial of degree at least $2$. Let $\Pi_f(x)$ denote the set of primes $p\leqslant x$ for which $f$ has at least one root modu…

DFI conjecture. As $|S(x)| \to \infty$, the probability that $\alpha \leq \frac{\nu}{p} < \beta$ approaches $\beta-\alpha$.

Let $n$ and $r$ be positive integers. Polynomial Bézout conjecture. There exist polynomials $A_{n,r}$ and $B_{n,r}$ in $q$ with integer coefficients such that … This conjecture con…