  For a given elliptic curve $E/\mathbb{Q}$, set $N_p(E)$ to 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 pairwise distinct primes
$p_1,\ldots,p_k$ of good reduction for $E$, for which 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 also conjecture that this result also holds for $k=1$ and $k=2$. In
particular for $k=1$ this conjecture says that there are "elliptic progressions
of primes" i.e. sequences of primes $p_1<p_2\cdots <p_m$ of arbitrary lengths
$m$ such that $N_{p_1}(E)=N_{p_2}(E)=\cdots =N_{p_m}(E)$.
