16 problems

Let $q=19^3$, let … and let $[5]=T^{q^5}-T$ and $M_5$ be Thakur's polynomial. The polynomial $P\in\mathbf{F}_{19^3}[T]$ is the degree-five $c$-Wieferich prime exhibited in the pape…

For a prime $p$, let $q_p(2)$ denote the Fermat quotient of $2$, and let $\log_{\mathcal{A}}(2)$ be its associated element of $\mathcal{A}$. A Wieferich prime is a prime satisfying…

Let $A$ be the polynomial ring in the paper, let $phi$ be the given Drinfeld module, and let $a\in A$ be the fixed base. A polynomial has no square factors when it is squarefree, a…

Let $d=p$ be an odd rational prime, let $K=\mathbb{Q}(\sqrt{p})$ be the corresponding real quadratic field, and write its fundamental unit as … with $\delta=1$ if…

Let $E/\mathbb{Q}$ be an elliptic curve and let $P\in E^0(\mathbb{Q})$ be a non-torsion point. Define the sets of good primes … and $W_{E,P}$ as the set of primes for which…

Let $\mathcal{P}$ be the set of all primes, and call a prime Wieferich if it satisfies the relevant congruence $N^{p-1}\equiv1\pmod{p^2}$ for the base under consideration. Wieferic…

Infinitude conjecture for $\mu$-hits. There are infinitely many $\mu$-hits.

Let $B_n$ be the balancing sequence defined by $B_0=0$, $B_1=1$, and $B_{n+1}=6B_n-B_{n-1}$ for $ngeq 1$. For a positive integer $m$, let the period of the balancing sequence modul…

Let $p<q$ be primes. A pair $(p,q)$ is called a Wieferich pair when it satisfies the paper's Wieferich condition. Wieferich-free two-prime $S$-Diophantine quadruple conjecture. If…

Let $p<q$ be primes. A pair $(p,q)$ is called a Wieferich pair when it satisfies the paper's Wieferich condition. Wieferich-free two-prime $S$-Diophantine quadruple conjecture. If…

Let $\Delta$ be a discriminant in case (B), and call a prime $p$ exceptional when it has the exceptional property defined in the paper. Exceptional-prime conjecture in case (B). Fo…

The Wieferich-prime counting asymptotic. For any integer $v\geq 2$, the elements $\\{x_p(v)\\}\in\mathcal{P}(v)$ for which $x_p(v)=0$ satisfy

Let $X=\{a_1,\ldots,a_m\}$ be a finite set of algebraic numbers, and let $K$ be a number field containing all $a_i$. An $X$-base Fibonacci-Wieferich prime is a prime ideal…

Wieferich prime conjecture. For any integer $w\ne 0$, there are infinitely many Wieferich primes to the base $w$. Moreover, if $w\notin\{1,-1\}$, then

Let $A=\mathbb{F}_q[T]$ have characteristic $p$, and let a Wieferich prime in $A$ mean a prime satisfying the Wieferich condition discussed in the paper. Thakur's conjecture. For e…

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…