There are some conjectures of the form: There always exist at least $X$ prime numbers between $A$ and $B$. Examples:

Bertrand's postulate: for every $n>1$ there is always at least one prime $p$ such that $n < p < 2n$.

Legendre's conjecture: there is a prime number between $n^2$ and $(n + 1)^2$ for every positive integer $n$.

Brocard's conjecture: there are at least four prime numbers between $p_{n}^2$ and $p_{n+1}^2$.

Oppermann's conjecture: there is at least one prime number between $n(n-1)$ and $n^2$.

If we denote by $\pi(x)$ the prime-counting function we can rewrite the above conjectures in the following form:

**Bertrand's postulate:**$\pi(2n)-\pi(n) \ge 1$ for $n>1$**Legendre's conjecture:**$\pi(n+1)^2)-\pi(n^2) \ge 1$**Brocard's conjecture:**$\pi(p_{n+1}^2)-\pi(p_{n}^2) \ge 4$**Oppermann's conjecture:**$\pi(n^2)-\pi(n(n-1)) \ge 1$I computed and saw that $f(n) = \pi(n^2)+\pi(n)+2-\pi((n+1)^2)$ is increasing when $n$ increasing and $f(n)\ge 0$ for all $n=1, 2, \dots, 18700$ (equivalent to $n^2=1, 4, 25 \cdots , 3.5\times 10^8)$.

Graph of $(n,f(n))$ where $f(n) = \pi(n^2)+\pi(n)-\pi((n+1)^2); \; 370 \le n \le 1.1\times10^4$

- So I proposed two conjecture as follows:

Conjecture 1:For every positive integer $n$, the number of primes between $n^2$ and $(n + 1)^2$ is less than the number of primes between $1$ and $n$ add $2$:$$\pi((n+1)^2)-\pi(n^2) \le \pi(n)+2.$$

Conjecture 2:For every positive integer $n$ greater than $369$, the number of primes between $n^2$ and $(n + 1)^2$ is less than the number of primes between $1$ and $n$:$$\pi((n+1)^2)-\pi(n^2) \le \pi(n).$$

Could you give a remark, comment, reference, or proof?

Noting that if the conjecture is true, it is stronger than a special case of the Second Hardy–Littlewood conjecture but this conjecture is not contradictory with the K-Tuple conjecture.

PS: In my computation I see that:

$$\lim_{n \to +large } \frac{\pi((n+1)^2)-\pi(n^2)}{\pi(n)}=1$$

What do You think with this equality?

a special caseof the second Hardy-Littlewood conjecture (the special case being $x=n^2$ and $y=2n+1$). BTW I believe that this conjecture (if true) is also out of reach. We do not know enough about the number of primes in such short intervals. $\endgroup$ – GH from MO Jul 5 '18 at 17:29not strongerthan the second Hardy-Littlewood conjecture (contrary what you say in your post). Instead, your conjecture is stronger than acertain special caseof the second Hardy-Littlewood conjecture. $\endgroup$ – GH from MO Jul 5 '18 at 17:58