[Legendre's conjecture](https://en.wikipedia.org/wiki/Legendre%27s_conjecture), proposed by Adrien-Marie Legendre, state that there is a prime number between $n^2$ and $(n+1)^2$ for every positive integer $n$.


My conjecture: Let $n$ be a positive integer, $p\_1$, $p\_2$ and $p\_3$ be odd prime numbers, and $n\ge2p\_1$, ($n\ge6$)


There is at least one prime number $p\_2$, so that $n^2 < p\_2 \le n^2 + n - p\_1$, and there is at least one prime number $p\_3$, so that $n^2 + n - p\_1 \le p\_3 < (n + 1)^2$



The equivalent proposition of Legendre's conjecture
Your conjecture for sufficiently large $n$ is implied by [Cramer's conjecture](https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture). In general though, conjectures like this unless they are coming from some specific application aren't that interesting. It is very easy to make many similar conjectures.


