Brocard's conjecture

Introduction In number theory, Brocard's conjecture is the conjecture that there are at least four prime numbers between (pn)2 and (pn+1)2, where pn 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 2025. Legendre's conjecture, which states that there is a prime between consecutive integer squares, directly implies that there are at least two primes between prime squares for pn ≥ 3 since pn+1 − pn ≥ 2.

Mathematical statement

Let be the -th prime, and let be the number of prime numbers . Formally, Brocard's conjecture claims:

This is equivalent to saying that there are at least four primes between squared consecutive primes other than and .

Relation to other open problems in mathematics

Legendre's conjecture Legendre's conjecture claims that there is a prime number between and for all natural number . It is an unsolved problem in mathematics as of 2025. If Legendre's conjecture is true, it immediately implies a weak version of Brocard's conjecture:

Cramér's conjecture Cramér's conjecture claims that , which gives a bound on how far apart primes can be. Cramér's conjecture implies Brocard's conjecture for sufficiently large .

Oppermann's conjecture Oppermann's conjecture claims that there is a prime in the interval and in the interval . This unsolved problem directly implies Brocard's conjecture.

We begin with the fact that , meaning that the minimal interval between primes is . Then, according to Oppermann's conjecture, there is a prime in the interval , a prime in the interval , a prime in the interval , and a prime in the interval . Then, we have:

Which implies at least primes between and , and because , there are at least primes between any two squared consecutive primes, which is exactly what Brocard's conjecture claims.

Examples n Prime numbers 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 stands for .

It is easy to verify the conjecture for small :

The number of primes between prime squares is 2, 5, 6, 15, 9, 22, 11, 27, ... . See the table (right) for a list of primes sorted by the difference. See the animation (right) for the first 30 differences.

Current research and results

Unconditional results

Bertrand's postulate A trivial result from Bertrand's postulate, a proven theorem, states that because there is a prime in the interval , and the length of the interval is much greater than , Bertrand's postulate suggests many primes in the interval , though not a sharp bound.

Baker-Harman-Pintz bound Using the bound proven by Baker et al., that , one can show that there exist infinitely many such that there is at least one prime in the interval , which is a much weaker result than Brocard's conjecture.

Conditional results

Legendre's Conjecture - weak version of Brocard's conjecture As shown above, Legendre's conjecture implies a weak version of Brocard's conjecture but is a strictly weaker conjecture.

Oppermann's Conjecture - full proof of Brocard's conjecture As shown above, Oppermann's conjecture directly implies Brocard's conjecture for large enough , which constitutes a proof of Brocard's conjecture.

Cramér's Conjecture - full proof of Brocard's conjecture As shown above, Cramér's conjecture implies Brocard's conjecture directly.

The Riemann Hypothesis - full proof of Brocard's conjecture The Riemann Hypothesis implies the bound , which implies Brocard's conjecture for sufficiently large , similarly to Cramér's conjecture.

Brocard's conjecture

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