The Brocard Conjecture and Primes in Small Intervals
Jan Feliksiak *
Edoo.pl Language School, Michaowski 11/8, 31-126 Krakow, Poland.
*Author to whom correspondence should be addressed.
Abstract
This paper presents the results of a research into the topic of the distribution of prime numbers in small intervals. The Brocard Conjecture asserts that the number of primes, within the interval, between the squares of two subsequent primes is:
\(\left(\pi_{p_{(n+1)}^{2}}-\pi_{p_{(n)}^{2}}\right) \geq 4 \quad \forall p_{n} \in \mathbb{N} \mid n \geq 2\) (0.1)
Although the number of primes within this interval varies to a great degree, there is a common ground, which makes it possible to settle this old conundrum. Three bounds are developed: the least lower bound and the lower/upper bounds. The least lower bound is implemented to prove the conjecture. The lower/upper bounds exploit the shortest such interval, namely between the twin primes. This has been done in order to establish the bounds, on the smallest number of primes within that interval. The research objective was not only to provide a true/false answer, but to clarify some aspects of the distribution of prime numbers within this interval as well. In addition to the weaker form of the Brocard Conjecture, the stronger one is also presented, as well as a couple others.
Keywords: Brocard conjecture, maximal prime gaps upper bound, prime number theorem, twin primes