# 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?

• Jan 4, 2022 at 2:08

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 $$k$$.