How would one solve the conjecture that for any odd n, there is a twin prime between $n^2$ and $(n+2)^2$?

Question

How would one solve the conjecture that for any odd n, there is a twin prime between $n^2$ and $(n+2)^2$?

Examples, for $n=3$, there is a twin prime between 9 and 25 of (11,13). For $n=9$, there is a twin prime between 81 and 121 of (101,103).

Answer 1

Out of curiosity, I wrote & ran a fairly simple program to check certain information regarding how many twin primes there are between $n^2$ and $\left(n + 2\right)^2$ for odd natural numbers $n$. I ran this for $n$ up to a million, i.e., $100000$, so the square goes up to a trillion. For each range of billion integers (apart from the last one due to a small coding limitation), I output the cumulative minimum, maximum & average of the # of twin primes in each $n^2$ to $\left(n + 2\right)^2$ range. The minimum value of $1$ first occurs for $n = 21$, but I didn't check if it occurred again afterwards. The maximum value goes up for each billion initially, but then sometimes doesn't change for a span of quite a few billion. At the end, it is $7207$. The average seems to always be increasing fairly steadily, but slower later on. At the end, it is about $3739.588515$.

This indicates your conjecture appears to be plausible, but it's obviously not a proof. As Mostafa Ayaz states, the twin prime conjecture is not proven so we don't even know for sure that there's an infinite # of twin primes, much less at least one between each $n^2$ and $\left(n + 2\right)^2$ range.

Answer 2

You can't.

According to Twin primes on Wikipedia, the number of twin prime couples is unknown (even whether they are finite or infinite). If your statement holds true, then as a consequence you have proved that there are infinitely many twin primes which is yet unsolved. Your question then sounds very encouraging as a rush to solve this old and interesting conjecture!