TITLE: Elliptic curves and prime numbers
QUESTION [6 upvotes]: 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 $\#E(F_{p_n})=p_{n+1}$?

REPLY [14 votes]: (Sorry, I misread the question at first.)  The following result reduces your question to a problem of analytic number theory:
Theorem (Hasse-Deuring-Waterhouse): For a prime $p$ and $N \geq 1$ the following are equivalent:
(i) There is an elliptic curve $E_{/\mathbb{F}_p}$ such that $\# E(\mathbb{F}_p) = N$.
(ii) We have $|N-(p+1)| \leq 2\sqrt{p}$.
As long as $p > 3$ (i.e., $p = p_n$ for $n \geq 3$) every elliptic curve can be put in "short Weierstrass form" $y^2 = x^3 + ax +b$.
So you are reduced to asking: is it true that for all $n > 3$ we have 
$|p_{n+1} - (p_n+1)| \leq 2 \sqrt{p_n}$?
According to this esteemed source, having this kind of upper bound on the prime gap (always) is conjectured to be true but far from being proven.  In fact, if I have it right this precisely Andrica's Conjecture.  As Dror Speiser points out, it is not even known conditionally on the Riemann Hypothesis.