number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, 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 or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

that there are at least four

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

s 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 Pierre René Jean Baptiste Henri Brocard (; 12 May 1845 – 16 January 1922) was a French meteorologist and mathematician, in particular a geometer. His best-known achievement is the invention and discovery of the properties of the Brocard p ...

. It is widely believed that this conjecture is true. However, it remains unproven as of 2025. The number of primes between prime squares is 2, 5, 6, 15, 9, 22, 11, 27, ... .

Legendre's conjecture Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between n^2 and (n+1)^2 for every positive integer n. The conjecture is one of Landau's problems (1912) on prime numbers, and is one of many open prob ...

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.

Notes

Conjectures about prime numbers Unsolved problems in number theory Squares in number theory {{Numtheory-stub

See also

Notes