Brocard's conjecture holds that there are at least four prime numbers between the square of the nth prime and the square of the (n+1)th prime, for every n of 2 or more. It is named after Henri Brocard and is widely believed true, but remains unproven. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementUsing the prime counting function, which gives the number of primes up to a given value, the conjecture claims that the count of primes up to the square of the (n+1)th prime, minus the count of primes up to the square of the nth prime, is at least four, for every n of two or more. Equivalently, there are at least four primes strictly between the squares of any two consecutive primes other than the primes two and three. 1 Progress Toward ResolutionNo unconditional proof of the conjecture is known. Legendre's conjecture, if proved, would only give a weaker version of the result. A full proof would follow from any of three other unproven statements: Oppermann's conjecture, Cramer's conjecture, or the Riemann Hypothesis, each of which implies Brocard's conjecture for sufficiently large n if it were itself proved. 1 Classification
Resolution Status Prize Status
Prize Status (category) Sources
1. Brocard's Conjecture (Wikipedia)
Wikimedia FoundationLead section
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
Mathematical statement section, opening sentence
Let pn be the nth prime, and let pi(x) be the number of prime numbers <= x. Formally, Brocard's conjecture claims:
Conditional results section, Oppermann's Conjecture subsection
As shown above, Oppermann's conjecture directly implies Brocard's conjecture for large enough n, which constitutes a proof of Brocard's conjecture.
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