Bertrand's postulate: Difference between revisions

Latest revision as of 22:28, 6 April 2009

Let $n$ be a natural number greater than $1$ . Then, there exists a prime number $p$ such that $n < p < 2 n$ .

In other words, the prime gap, i.e., the gap between a prime $p$ and the next prime, is strictly smaller than $p$ .

Cramér's prime gap conjecture states that the prime gap (i.e., the gap between a prime and the next prime) is $O ((\log p)^{2})$ .
The prime-between-squares conjecture states that there exists a prime between the squares of any two consecutive natural numbers.
The Riemann hypothesis implies the large prime gap conjecture, which states that the prime gap is $O (\sqrt{p} \log (p))$ .

Rankin's bound states that there exist arbitrarily large primes $p$ for which the prime gap is:

$Ω (\log p \frac{\log \log p \log \log \log \log p}{(\log \log \log p)^{2}}$ .

@@ Line 5: / Line 5: @@
 Let <math>n</math> be a natural number greater than <math>1</math>. Then, there exists a [[prime number]] <math>p</math> such that <math>n < p < 2n</math>.
-In other words, the [[[prime gap]], i.e., the gap between a prime <math>p</math> and the next prime, is strictly smaller than <math>p</math>.
+In other words, the [[prime gap]], i.e., the gap between a prime <math>p</math> and the next prime, is strictly smaller than <math>p</math>.
+==Relation with other facts and conjectures==
+===Upper bounds on the limit superior of prime gap===
+* [[Cramér's prime gap conjecture]] states that the [[prime gap]] (i.e., the gap between a prime and the next prime) is <math>O((\log p)^2)</math>.
+* The [[prime-between-squares conjecture]] states that there exists a prime between the squares of any two consecutive natural numbers.
+* The [[Riemann hypothesis]] implies the [[large prime gap conjecture]], which states that the prime gap is <math>O(\sqrt{p}\log(p))</math>.
+===Lower bounds on the limit superior of prime gaps===
+* [[Rankin's bound]] states that there exist arbitrarily large primes <math>p</math> for which the prime gap is:
+<math>\Omega(\log p \frac{\log \log p \log \log \log \log p}{(\log \log \log p)^2}</math>.