Brocard's conjecture
# Brocard's conjecture

In number theory, Brocard's conjecture is the conjecture that there are at least four prime numbers between $(p_n)^2$ and $(p_{n+1})^2,$ where $p_n$ is the nth prime number, for every n ≥ 2. The conjecture is named after Henri Brocard. It is widely believed that this conjecture is true. However, it remains unproven as of 2022.

| n | $p_{n}$ | $p_{n}^{2}$ | Prime numbers | $\Delta$ |
| - | - | - | - | - |
| 1 | 2 | 4 | 5, 7 | 2 |
| 2 | 3 | 9 | 11, 13, 17, 19, 23 | 5 |
| 3 | 5 | 25 | 29, 31, 37, 41, 43, 47 | 6 |
| 4 | 7 | 49 | 53, 59, 61, 67, 71... | 15 |
| 5 | 11 | 121 | 127, 131, 137, 139, 149... | 9 |
| $\Delta$ stands for $\pi (p_{n+1}^{2})-\pi (p_{n}^{2})$. |||||

The number of primes between prime squares is 2, 5, 6, 15, 9, 22, 11, 27, ... OEIS: A050216.

Legendre's conjecture that there is a prime between consecutive integer squares directly implies that there are at least two primes between prime squares for $p_n$ ≥ 3 since $p_{n+1}$ − $p_n$ ≥ 2.

