In light of the recent work by Maynard and Tao on the Dickson $k$-tuples conjecture, we show that with a small improvement in the known bounds for this conjecture, we would be able to prove that for some fixed $R$, there are infinitely many Carmichael numbers with exactly $R$ factors for some fixed $R$. In fact, we show that there are infinitely many such $R$.

Let $E$ be an elliptic curve over $\mathbb{Q}$ without complex multiplication, and for each prime $p$ of good reduction, let ${{n}_{E}}(p)\,=\,\left| E\left( {{\mathbb{F}}_{p}} \right) \right|$. For any integer $b$, we consider elliptic pseudoprimes to the base $b$. More precisely, let ${{Q}_{E,B}}(x)$ be the number of primes $p\le x$ such that ${{b}^{{{n}_{E}}(p)}}\,\equiv \,b\left( \bmod \,{{n}_{E}}\left( p \right) \right)$, and let $\pi _{E,b}^{\text{pseu}}(x)$ be the number of compositive${{n}_{E}}(p)$ such that ${{b}^{{{n}_{E}}(p)}}\,\equiv \,b\left( \bmod \,{{n}_{E}}\left( p \right) \right)$ (also called elliptic curve pseudoprimes). Motivated by cryptography applications, we address the problem of finding upper bounds for ${{Q}_{E,B}}(x)$ and $\pi _{E,b}^{\text{pseu}}(x)$, generalising some of the literature for the classical pseudoprimes to this new setting.