In
number theory, Brocard's conjecture is the
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
that there are at least four
prime numbers between (''p''
''n'')
2 and (''p''
''n''+1)
2, where ''p''
''n'' is the ''n''
th 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.
The number of primes between prime squares is 2, 5, 6, 15, 9, 22, 11, 27, ... .
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.
See also
*
Prime-counting function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ).
History
Of great interest in number theory is t ...
Notes
Conjectures about prime numbers
Unsolved problems in number theory
Squares in number theory
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