6 problems

Let $v_p$ denote the $p$-adic valuation. The variables $x$ and $q$ are integers with $x>1$, and $p$ and $q$ are primes in the second assertion. Sharp upper-bound conjecture for Fer…

Let $p$ range over primes, let $v_p$ denote the $p$-adic valuation, and fix an integer $x>1$. Finiteness conjecture for exceptional Fermat quotients. The inequality … occurs for on…

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…

For a prime $p>2$, write $q_2(p)=(2^{p-1}-1)/2$. Arithmetic progression nonvanishing conjecture. For every pair of positive integers $m>a>0$ with $\gcd(m,a)=1$, there are infinitel…

Uniformity and independence conjecture. As $s$ varies over ${\mathcal S}_p$, the values $A_{p,s}$ behave uniformly randomly modulo $p$, and the values for the various $s$ are indep…

Skula's congruence. One has