Recall that Legendre's Conjecture states that for every positive integer $n$, there is always a prime $p$ such that $n^2 < p < (n + 1)^2$.
I am currently working on a symmetrical approach to this problem. Instead of analyzing every square, I anchor the search exclusively on even squares to leverage parity and additive symmetries. I am investigating the following two complementary hypotheses:
The Upper Gap: For every even integer $n \ge 2$, there exists a prime $q < 2n + 1$ such that $n^2 + q$ is prime.
The Lower Gap: For every even integer $n > 2$, there exists a prime $q < 2n - 1$ such that $n^2 - q$ is prime.
Because every odd square is sandwiched exactly between two even squares, satisfying both conditions for all even $n$ guarantees a prime in every gap between consecutive squares, effectively implying Legendre's Conjecture. To justify the tight bounding constraints ($q < 2n \pm 1$), I used a standard Cramér-style probabilistic heuristic based on the Prime Number Theorem. The expected number of successful prime offsets $q$ within the bounds is roughly:$$E(n) \approx \frac{n}{\ln(2n)\ln(n)}$$Since $E(n)$ diverges rapidly, the heuristic suggests these bounded primes should always exist. I have computationally verified both conditions up to $n = 10^8$. At this limit, the theoretical upper bound for $q$ is $200,000,001$, but the actual maximum required prime $q$ for the upper gap was remarkably small, at just $q = 2,797$.
I am new to all these sites like MathStackexchange and MathOverflow, I just observed this interesting pattern that makes a stronger claim than Legendre's conjecture and wanted to share it, my main question would be if this provides any insight or potential approaches into proving Legendre's conjecture, and what would be a good threshold to test this conjecture up to (I recognize that $10^8$ is small in the world of primes.)
I asked this question in MathStackexchange and I wanted to ask it here because I think I can receive a better answer regarding my mathematical inquiry, as for the computational side, it does not matter as much to me (it is just a reinforcement of what I believe to be true.)
Thanks for your time!
