Editing Bertrand's postulate (section)

== Better results ==

It follows from the prime number theorem that for any [[real number|real]] <math>\varepsilon > 0</math> there is a <math>n_0 > 0 </math> such that for all <math>n > n_0</math> there is a prime <math>p</math> such that <math>n < p < (1 + \varepsilon) n</math>. It can be shown, for instance, that

:<math>\lim_{n \to \infty}\frac{\pi((1+\varepsilon)n)-\pi(n)}{n/\log n} = \varepsilon,</math>

which implies that <math> \pi (( 1 + \varepsilon ) n) - \pi (n) </math> goes to infinity (and, in particular, is greater than 1 for [[sufficiently large]] <math>n</math>).<ref>G. H. Hardy and E. M. Wright, ''An Introduction to the Theory of Numbers'', 6th ed., Oxford University Press, 2008, p. 494.</ref>

Non-asymptotic bounds have also been proved. In 1952, Jitsuro Nagura proved that for <math>n \ge 25</math> there is always a prime between <math>n</math> and <math>\bigl(1+\tfrac{1}{5} \bigr) n</math>.<ref>{{Citation | last1 = Nagura | first1 = J | year = 1952 | title = On the interval containing at least one prime number | url = http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.pja/1195570997&view=body&content-type=pdf_1 | journal = Proceedings of the Japan Academy, Series A | volume = 28 | issue = 4| pages = 177–181 | doi=10.3792/pja/1195570997| doi-access = free }}</ref>

In 1976, [[Lowell Schoenfeld]] showed that for <math>n \ge 2\,010\,760</math>, there is always a prime <math>p</math> in the [[open interval]] <math>n < p < \bigl(1+\tfrac{1}{16\,597} \bigr) n</math>.<ref>{{Citation|author=Lowell Schoenfeld|title=Sharper Bounds for the Chebyshev Functions ''θ''(''x'') and ''ψ''(''x''), II|journal=Mathematics of Computation|volume=30|issue=134|pages=337–360|date=April 1976|doi=10.2307/2005976|jstor=2005976}}</ref>

In his 1998 doctoral thesis, [[Pierre Dusart]] improved the above result, showing that for <math>k \ge 463</math>, 
<math>p_{k+1} \le \left( 1 + \frac{1}{2 \log^2{p_k}} \right) p_k</math>,
and in particular for <math>x \ge 3\,275</math>, there exists a prime <math>p</math> in the interval <math>x < p \le \left( 1 + \frac{1}{ 2 \log^2{x} } \right) x</math>.<ref>{{Citation | last = Dusart | first = Pierre | author-link = Pierre Dusart | url = http://www.unilim.fr/laco/theses/1998/T1998_01.pdf | title = Autour de la fonction qui compte le nombre de nombres premiers | type= PhD thesis | language = french | year = 1998 }}</ref>

In 2010 Pierre Dusart proved that for <math>x \ge 396\,738</math> there is at least one prime <math>p</math> in the interval <math>x < p \le \left( 1 + \frac{1}{ 25 \log^2{x} } \right) x</math>.<ref>{{cite arXiv | last1 = Dusart | first1 = Pierre | author-link1 = Pierre Dusart | eprint = 1002.0442 | year = 2010 | title = Estimates of Some Functions Over Primes without R.H.| class = math.NT }}</ref>

In 2016, Pierre Dusart improved his result from 2010, showing (Proposition 5.4) that if <math>x \ge 89\,693</math>, there is at least one prime <math>p</math> in the interval <math>x < p \le \left( 1 + \frac{1}{ \log^3{x} } \right) x</math>.<ref>{{Citation|journal= The Ramanujan Journal | volume = 45 | pages = 227–251 | last1 = Dusart | first1 = Pierre | author-link1 = Pierre Dusart | year = 2016 | title = Explicit estimates of some functions over primes | doi=10.1007/s11139-016-9839-4| s2cid = 125120533 }}</ref> He also shows (Corollary 5.5) that for <math>x \ge 468\,991\,632</math>, there is at least one prime <math>p</math> in the interval <math>x < p \le \left( 1 + \frac{1}{ 5\,000 \log^2{x} } \right) x</math>.

Baker, Harman and Pintz proved that there is a prime in the interval <math>[x-x^{0.525},\,x]</math> for all sufficiently large <math>x</math>.<ref name="baker">{{Citation |last1=Baker |first1=R. C. |first2=G. |last2=Harman |first3=J. |last3=Pintz |year=2001 |title=The difference between consecutive primes, II |journal=Proceedings of the London Mathematical Society |volume=83 |issue=3 |pages=532–562 |doi=10.1112/plms/83.3.532 |citeseerx=10.1.1.360.3671 |s2cid=8964027 }}</ref>

Dudek proved that for all <math>n \ge e^{e^{33.3}}</math>, there is at least one prime between <math>n^3</math> and <math>(n + 1)^3</math>.<ref>{{Citation | last = Dudek | first = Adrian | title = An explicit result for primes between cubes | journal = Funct. Approx. | volume = 55 | issue = 2 | date = December 2016 | pages = 177–197 | doi = 10.7169/facm/2016.55.2.3 | arxiv = 1401.4233 | s2cid = 119143089 }}</ref>

Dudek also proved that the [[Riemann hypothesis]] implies that for all <math>x \geq 2</math> there is a prime <math>p</math> satisfying
:<math>x - \frac{4}{\pi} \sqrt{x} \log x < p \leq x.</math><ref>{{Citation| last=Dudek|first=Adrian W.|date=21 August 2014|title=On the Riemann hypothesis and the difference between primes|journal=International Journal of Number Theory|volume=11|issue=3|pages=771–778|doi=10.1142/S1793042115500426|issn=1793-0421|arxiv=1402.6417|bibcode=2014arXiv1402.6417D|s2cid=119321107}}</ref>