TITLE: Are there arbitrarily long prime deserts?
QUESTION [6 upvotes]: Possible Duplicate:
What is the maximum number of consecutive composite numbers possible? 

Define a prime desert of length $k$ to be a sequence of numbers $n + 1, n + 2, ..., n + k $ such that $n + i$ is composite for $1 \le i \le k$.  So my question is given a positive integer $k$ is there a prime desert of length $k$?

REPLY [5 votes]: Here's an alternative approach that uses some heavier machinery.
We know that, if $\pi(x)$ denotes the number of primes $\leq x$, then $\pi(x)$ grows at roughly the same rate as $\frac{x}{\ln x}$, in the sense that $\lim_{x\to\infty} \frac{\pi(x)}{x/\ln (x)}=1$. (This statement is called The Prime Number Theorem.) Now, if the largest prime desert were of size $k$, then $\pi(x)\geq\frac{x}{k+1}$. But $\lim_{x\to\infty}\frac{x/(k+1)}{x/\ln(x)}=\lim_{x\to\infty}\frac{\ln(x)}{k+1}=\infty$.