Let $p_n$ be the $n^{th}$ prime number. Suppose $E(F_{p_n})$ denotes an elliptic curve over the Galois field $GF(p_n)$ which is defined by $y^2=x^3+ax+b$. Is the below claim true?

For each integer number $n>3$, there exist integer numbers $a$ and $b$ such that $\#E(F_{p_n})=p_{n+1}$?