Editing Primorial (section)

== Characteristics ==

* Let {{mvar|p}} and {{mvar|q}} be two adjacent prime numbers. Given any <math>n \in \mathbb{N}</math>, where <math>p\leq n<q</math>:
:<math>n\#=p\#</math>
* The fact that the [[binomial coefficient]] <math>\tbinom{2n}{n}</math> is divisible by every prime between <math>n+1</math> and <math>2n</math>, together with the inequality <math>\tbinom{2n}{n} \leq 2^{n}</math>, allows to derive the upper bound:<ref>G. H. Hardy, E. M. Wright: ''An Introduction to the Theory of Numbers''. 4th Edition. Oxford University Press, Oxford 1975. {{ISBN|0-19-853310-1}}.<br />Theorem 415, p.&nbsp;341</ref>
:<math>n\#\leq 4^n</math>.

Notes:

# Using elementary methods, mathematician Denis Hanson showed that <math>n\#\leq 3^n</math><ref>{{Cite journal |last=Hanson |first=Denis |date=March 1972 |title=On the Product of the Primes |journal=[[Canadian Mathematical Bulletin]] |volume=15 |issue=1 |pages=33–37 |doi=10.4153/cmb-1972-007-7|doi-access=free |issn=0008-4395}}</ref>
# Using more advanced methods, Rosser and Schoenfeld  showed that <math>n\#\leq (2.763)^n</math><ref name="RosserSchoenfeld1962">{{Cite journal |last1=Rosser |first1=J. Barkley |last2=Schoenfeld |first2=Lowell |date=1962-03-01 |title=Approximate formulas for some functions of prime numbers |journal=Illinois Journal of Mathematics |volume=6 |issue=1 |doi=10.1215/ijm/1255631807 |issn=0019-2082|doi-access=free }}</ref>
# Rosser and Schoenfeld in Theorem 4, formula 3.14, showed that for <math>n \ge 563</math>, <math>n\#\geq (2.22)^n</math><ref name="RosserSchoenfeld1962"/>

* Furthermore:
:<math>\lim_{n \to \infty}\sqrt[n]{n\#} = e </math>
:For <math>n<10^{11}</math>, the values are smaller than [[e (mathematical constant)|{{mvar|e}}]],<ref>L. Schoenfeld: ''Sharper bounds for the Chebyshev functions <math>\theta(x)</math> and <math>\psi(x)</math>''. II. ''Math. Comp.'' Vol.&nbsp;34, No.&nbsp;134 (1976) 337–360; p.&nbsp;359.<br />Cited in: G. Robin: ''Estimation de la fonction de Tchebychef <math>\theta</math> sur le {{mvar|k}}-ieme nombre premier et grandes valeurs de la fonction <math>\omega(n)</math>, nombre de diviseurs premiers de {{mvar|n}}''. ''Acta Arithm.'' XLII (1983) 367–389 ([http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4242.pdf PDF 731KB]); p.&nbsp;371</ref> but for larger {{mvar|n}}, the values of the function exceed the limit {{mvar|e}} and oscillate infinitely around {{mvar|e}} later on.

* Let <math>p_k</math> be the {{mvar|k}}-th prime, then <math>p_k\#</math> has exactly <math>2^k</math> divisors. For example, <math>2\#</math> has 2 divisors, <math>3\#</math> has 4 divisors, <math>5\#</math> has 8 divisors and <math>97\#</math> already has <math>2^{25}</math> divisors, as 97 is the 25th prime.
* The sum of the reciprocal values of the primorial [[Convergent series|converges]] towards a constant
:<math>\sum_{p\,\in \,\mathbb{P}}  {1 \over p\#} = {1 \over 2} + {1 \over 6} + {1 \over 30} + \ldots = 0{.}7052301717918\ldots</math>
:The [[Engel expansion]] of this number results in the sequence of the prime numbers (See {{OEIS|A064648}})

* Euclid's proof of his [[Euclid's theorem|theorem on the infinitude of primes]] can be paraphrased by saying that, for any prime <math>p</math>, the number <math>p\# +1</math> has a prime divisor not contained in the set of primes less than or equal to <math>p</math>.