Brocard's conjecture
Brocard's conjecture states that for every prime p greater than 2, at least four primes lie strictly between p² and the square of the next prime.
Prime number: A natural number greater than 1 whose only positive divisors are 1 and itself. The conjecture counts primes inside each interval.
Prime-counting inequality: An inequality that bounds the number of primes in a specified range using the prime-counting function. Brocard's claim becomes a lower bound of four for the count between consecutive prime squares.
Henri Brocard: Henri Brocard was a French mathematician known for work in geometry and number theory. Brocard stated the conjecture that now bears his name.
Primes between 4 and 9: The primes strictly between 4 and 9 are 5 and 7. This interval corresponds to the first eligible prime, 2, but is excluded from the conjecture.
Consecutive primes: Two primes are consecutive when no prime lies strictly between them. Each interval runs from one prime's square to the next prime's square.
Bertrand's postulate: A theorem stating that for every integer n greater than 1, a prime lies between n and 2n. It guarantees one prime in a shorter multiplicative range, while Brocard's conjecture demands four in a different interval.
Brocard's 1904 paper: A paper by Henri Brocard that presented the assertion about primes between consecutive prime squares. It is the historical source associated with the conjecture's formulation.
Primes between 9 and 25: The primes strictly between 9 and 25 are 11, 13, 17, 19, and 23. The interval from 3² to 5² contains five primes, exceeding the conjectured minimum.
Prime gap: The difference between a prime and the next larger prime. The conjecture asks how many primes occur across a gap between squared endpoints.
Andrica's conjecture: A conjecture that the square roots of consecutive primes differ by less than 1. Both conjectures constrain consecutive prime gaps, but in distinct forms.