Runs of composite integers and the Chinese Remainder Theorem

Abstract

The distribution of the prime numbers is of course a well-trodden path. The casual enquirer, working with small numbers, soon finds that the distribution has no obvious regularities, and sees that the primes thin out. The Prime number theorem (which is not elementary) helps to firm up this observation: Prime number theorem: If π(n) is the number of primes no greater than n, then i.e. π(n)/(n/ln(n)) → 1 as n → ∞. Thus for the proportion of prime numbers upto to n, π(n)/n, we have that which tends slowly to 0.