On the equation $N_{p_1}(E)\cdot N_{p_2}(E)\cdots N_{p_k}(E)=n$
Kirti Joshi
TL;DR
This work investigates the distribution of the number of points $N_p(E)$ on elliptic curves modulo primes of good reduction, introducing $G_k(E,n)$ to count $k$-tuples of primes with a fixed product $N_p(E)$. Under GRH for non-CM curves (unconditionally for CM), it proves that $G_k(E,n)$ is unbounded for all $k\ge3$, and conjectures the same for $k=1,2$—the existence of arbitrarily long elliptic progressions of primes with equal $N_p(E)$. The approach blends division-field density, high-moment estimates via a fourth-moment calculation, and sieve techniques to transfer Erdős-type combinatorics to the elliptic-curve setting, with CM vs non-CM distinctions influencing the analytic tools used. The paper further extends the framework to the classical Erdős problem $(p_1-1)\cdots(p_k-1)=n$, providing analogous unboundedness results and stronger lower bounds for $k=2$ and $k\ge3$, and presents extensive numerical data illustrating elliptic progressions, highlighting notably richer phenomena in CM cases.
Abstract
For a given elliptic curve $E/\mathbb{Q}$, let $N_p(E)$ be the number of points on $E$ modulo $p$ for a prime of good reduction for $E$. Given integer $n$, let $G_k(E,n)$ be the number of $k$-tuples of $p_1<p_2<\ldots <p_k$ primes of good reduction for $E$, for which the equation in the title holds, then on assuming the Generalized Riemann Hypothesis for elliptic curves without CM (and unconditionally if the curves have complex multiplication), I show that $\varlimsup_{n\to\infty} G_k(E,n)=\infty$ for any integer $k\geq 3$. I conjecture that this result also holds for $k=1,2$ i.e. this conjecture says that there are arbitrarily long ``elliptic progressions of primes'' i.e. sequences of primes $p_1<p_2<\cdots <p_m$ of arbitrary lengths $m$ such that $N_{p_1}(E)=N_{p_2}(E)=\cdots =N_{p_m}(E)$.