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The prime gap $g_n$ is the difference $p_{n+1}-p_n$ between consecutive prime numbers. For the primes $2,3,5,7,11$, the first gaps are $1,2,2,4$. Every gap after the first is even because both endpoint primes are odd.

Prime gaps are unbounded: for any $m\ge 2$, the consecutive integers $m!+2,\dots ,m!+m$ are all composite, since each is divisible by its corresponding summand. Thus there are arbitrarily long prime-free intervals. This elementary argument proves existence without determining how large gaps typically are near a specified bound.

The prime index of a prime number $p$ is its position in the increasing sequence $p_1=2,p_2=3,p_3=5,\dots$. Thus the index of $11$ is $5$, and the prime-counting function satisfies $\pi (p_n)=n$.

Indexing begins at one here. An implementation using zero-based array positions must subtract one. The index describes ordering among primes, whereas a prime gap measures the numerical distance between neighboring primes. For nonprime $x$, $\pi (x)$ still counts primes up to $x$, but $x$ itself has no prime index.