If ratio: number of integers in between twin primes to number of all integers approaches but never reaches zero, does it imply infinite twin primes? [closed]

Solution 1:

There is a notion of 'natural density' : given a set (possibly infinite) $S$ of natural numbers (we don't really want 'integers' here because we're only considering the positive ones), we can look at the quantity $d(S)=\lim\limits_{n\to\infty}\dfrac{S_n}{n}$ where $S_n$ is the number of members of $S$ less than or equal to $n$. This limit won't always exist; if it does exist, though, it can be a useful quantity. For instance, $d(S\cup T) = d(S)+d(T)-d(S\cap T)$ if these limits all exist, and if $T\subset S)$ and both limits exist, then $d(T)\leq d(S)$. In particular, if $d(S)=0$ and $T\subset S$ then we know that $d(T)$ exists and is equal to $0$.

But knowing that $d(S)$ is zero doesn't give enough information to say that $S$ is infinite or finite. On the one hand, the density of the set of primes is zero; you can use a construction very similar to the one you use for twin primes, and by the statement above this immediately implies that the density of the set of twin primes is zero. On the other hand, the density of the set $S=\{1,2,3,4,5\}$ is also zero, because $S_n=5$ for $n\geq 5$ and so $d(S)=\lim\limits_{n\to\infty}\frac5n=0$.