Is it true that for any positive integer $n$, there exists an integer $x$ where there are at least $n$ primes between $x^2$ and $(x+1)^2$

Question

Am I correct that this follows directly from two observations:

(1) The sum of the reciprocals of primes diverges.

(2) The sum of the reciprocals of squares converges

Here's my thinking:

If there existed an integer $m$ that was a maximum number of primes between two consecutive squares, then the sum of the reciprocal of primes would be less than $m\frac{\pi^2}{6}$ which would be convergent which is not the case.

Am I correctly understanding the implication of these two observations? Are there other well known properties that stem from these two observations?

Answer 1

Your reasoning is mostly correct. Two slight imprecisions:

• Where it says “sum of the primes” it should say “sum of the reciprocals of the primes”, as above.
• It doesn’t make much sense to say “the sum would be less than $m\frac{\pi^2}6$ which would be convergent”, since $m\frac{\pi^2}6$ is a constant. The argument should be that the sum of the reciprocals of the primes would be dominated by the sum of the $m$-fold reciprocals of the squares, and would thus converge; its value would then be less than $m\frac{\pi^2}6$.

There’s no shortage of well-known properties that stem from $(1)$, but I’m not aware of any that stem from the combination of $(1)$ and $(2)$.