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Brocard's conjecture

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Brocard's conjecture pertains to the squares of prime numbers. Here we denote the th prime as .

Conjecture. (Henri Brocard) With the exception of 4, there are always at least four primes between the square of a prime and the square of the next prime. In terms of the prime counting function, this would mean that for all .

There are just two primes between 4 and 9, namely 5 and 7. But there are five primes between 9 and 25, and six between 25 and 49. If we write down the primes and the squares of primes in order (A000430) and then put the squares of primes in bold,

2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 289, 293, 307, ...

we see that the numbers in bold seem to get fewer and further apart. The conjecture has been has been checked up to .

A natural strengthening of the conjecture is that there are always at least four primes between and for . This conjecture has been checked to at least .

See also Legendre's conjecture.