Prime gaps are the difference between adjacent prime numbers. There is no function in n that can tell you what the $nth$ prime number is. But a lot can be infered from the properties of prime numbers. For example, all prime numbers except 2 are odd. Therefore the prime gaps must be even numbers. It was Euclid who provided this marvellous proof that there are an infinite number of prime numbers.

Suppose to the contrary that there are only a finite number of prime numbers. Therefore there must be a largest prime number, $p$. In that case $1 \times 2 \times 3 \cdots \times p + 1$ is either a prime bigger than $p$ or is divisibe by a prime bigger than $p$. In either case we have a contradiction. Ergo, reducto ad absurdum there is no greatest prime. Which implies there are an infinite number of prime numbers.

But what of the spacing between prime numbers? Is there two prime numbers that are separated by a space greater than a give integer? In effect we are asking whether there are a, say $k$, number of consecutive composite numbers. We see that $n!+2$, $n!+3$, …, $n!+n$ are $n-1$ consecutive composite numbers. So not only are there an infinite number of prime numbers, the gap between adjacent primes can be as large as necessary. That is, if you choose any integer $k$ then there are always two adjacent prime numbers that are separated by a gap larger than $k$.