3 problems

Let $n$ be an odd squarefree composite integer, and let Euler liars in $\mathbb{Z}_n^*$ mean the elements $a$ satisfying the Euler probable-prime congruence…

Let $M\geq 7$. Choose distinct primes $p,q$ uniformly at random subject to $5\leq p,q\leq M$, and set $N=pq$. Choose an elliptic curve $E(\mathbb{Z}/N\mathbb{Z})$ uniformly at rand…

Let the three propositions above be the results concerning the sets and quantities introduced in this section, each currently stated for a restricted range of $calpha$. Exponent co…