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 \(p_1<p_2<\cdots <p_k\) 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 \(\varlimsup _{n\rightarrow \infty } G_k(E,n)=\infty \) for any integer \(k\ge 3\) . 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 \(N_{p_1}(E)=N_{p_2}(E)=\cdots =N_{p_m}(E)\) .
