What is the ratio of prime numbers to perfect squares

I couldn't find this exact question. I know that there are an infinite number of prime numbers and positive squares. I also found that there are more prime numbers than perfect squares, but does the ratio of primes to perfect positive squares approach infinity or some other value?

Solution 1:

The number of squares in $[1, x]$ is asymptotically $\sqrt{x}$, whereas the number of primes in $[1,x]$ is asymptotically $x/\ln x$ by the prime number theorem, so you could say that in ratio there are more primes than squares, as $(\sqrt x)/(x/\ln x) \to 0$.

This makes sense if you think that probabilistically, the chance that a large given $N$ is prime (resp. square) is approximately $1/\ln N$ (resp. $1/\sqrt N$), and $1/\ln N > 1/\sqrt N$ for large $N$. That is, it is "more likely" for a large $N$ to be prime than square.

Edit: to more directly answer your original question, the above observations imply that the ratio of primes to squares in $[1,x]$ is asymptotically $\sqrt x/\ln x$, which goes to infinity. One can interpret this as "for sufficiently large $x$, there are more than $k$ times as many primes in $[1,x]$ as squares", and it will be true for any fixed $k$.