This question was so well stated by someone else that I am quoting their words here:
Let $a(n)$ be the number of pairs of twin primes between $n^2$ and $(n+1)^2$. Of course, if the twin primes conjecture is false then $a(n)$ is zero for large $n$. But is anything known or conjectured about the behaviour of $a(n)$ as $n→∞$?
For example, is it known or conjectured whether $a(n)$ is bounded? Or whether it tends to infinity?
Here's the work that I did up to now:
I noticed that there are always twin primes between $n^2$ and $(n+1)^2$ except for $n \in \left\{9,19,26,27,30,34,39,49,53,77,122\right\}$.
For $n >1330, a(n)$ fluctuates but continues to reach higher numbers and for $n>1330$ does not fall below $\left\lfloor\dfrac{\text{highest found so far}}{3}\right\rfloor$ for any $n$ that I checked.
For example:
- For $1000 \le n \le 2000$, $7 \le a(n) \le 35$
- For $4000 \le n \le 5000$, $17 \le a(n) \le 60$
- For $9000 \le n \le 10,000$, $30 \le a(n) \le 99$
- For $59,000 \le n \le 60,000$, $125 \le a(n) \le 379$
I suspect $a(n)$ must stop reaching higher numbers at some point because the sum of the reciprocals of twin primes is convergent. Does anyone know at what point it stops? Can anyone provide references to methods of analysis that can be used to determine this point.
In an answer to a similar question, the following analysis was given:
Suppose you have twin primes ($p_k+2=p_{k+1}$). Then the gap is roughly $4\sqrt{x}$ with numbers around $x$. Heuristics suggest that, on average, such an interval would contain about
$\frac{8C_2\sqrt{x}}{\log^2 x}$ twin primes
where $C_2\approx0.6601618158$ is the twin prime constant.
If we treat the primes as being Poisson distributed, the chance that no primes would be found in the interval is:
$\exp\left(-\frac{8C_2\sqrt{x}}{\log^2 x}\right)$
I am not clear on the details of this analysis. Can it be used to determine the point where $a(n)$ stops increasing.
Edit: Removed the term "pattern" from the question and removed the terms "maximum" and "minimum" which were used incorrectly. I reworded the question to focus on my main point.
Edit 2: I changed "number" to "frequency" based on a comment which is more accurate and changed my question to "at what point" as suggested by a comment. Thanks to everyone for all the comments.
The number of twin-primes between successive squares should increase. There will be some fluctuations yet there will be a very clear upwards trend.
Of course, this is conjecturally as no one knows if there are infinitely many twin-primes, so the count theoretically could be $0$ from some point on, but this is very unlikely.
In more detail: it is expected that the number of twin-prime pairs up to $x$ is asymptotically $$2C_2 \frac{x}{(\log x)^2}$$ where $C_2$ is the so-called twin-prime constant which is about $0.66$. This can be put differently as saying that a number of size about $x$, is twin-prime (say, the smaller of a pair for definiteness) with "probability" about
$$ \frac{2C_2}{(\log x)^2}.$$ Still differently an interval $[x, x+y]$ should contain $$ \frac{2C_2y}{(\log (x +y))^2}$$ twin-prime numbers if the $y$ is not too small relative to $x$; but it being of size $x^{\theta}$ for some positive $\theta$ should suffice. Again, this is of course conjecturally; it is not even known for the primes if there is prime in every interval $[x,x+\sqrt{x}]$ (this is Legendre's conjecture, though one is close).
So what does this tell us about the question: between $N^2$ and $(N+ 1)^2 = N^2 + 2N + 1$ we thus expect about $$\frac{2C_2 2N}{(\log (N+1)^2)^2}$$
which reduces to about $$\frac{C_2 N}{(\log N)^2}$$ which grows as $N$ grows.
This does not match the data you give too well, but first the numbers considered are small in that context and you only mention the maximum while what I give should be the typical/average value.