Editing Formula for primes (section)

==Mills' formula==
The first such formula known was established by {{harvs|first=W. H.|last=Mills|year=1947|txt}}, who proved that there exists a [[real number]] ''A'' such that, if

:<math>d_n = A^{3^{n}}</math>

then

:<math>\left \lfloor d_n \right \rfloor = \left \lfloor A^{3^{n}} \right \rfloor</math>

is a prime number for all positive integers <math>n</math>.<ref>{{citation|first=W. H.|last=Mills|title=A prime-representing function|journal=[[Bulletin of the American Mathematical Society]]|volume=53|year=1947|page=604|doi=10.1090/S0002-9904-1947-08849-2|issue=6|url = https://www.ams.org/journals/bull/1947-53-06/S0002-9904-1947-08849-2/S0002-9904-1947-08849-2.pdf|doi-access=free}}.</ref> If the [[Riemann hypothesis]] is true, then the smallest such <math>A</math> has a value of around 1.3063778838630806904686144926... {{OEIS|id=A051021}} and is known as [[Mills' constant]].<ref>{{citation|first1=Chris K.|last1=Caldwell|first2=Yuanyou|last2=Cheng|title=Determining Mills' Constant and a Note on Honaker's Problem|journal=Journal of Integer Sequences|volume=8|year=2005|at=Article 05.4.1.|url=https://cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.html}}</ref> This value gives rise to the primes <math>\left \lfloor d_1 \right \rfloor = 2</math>, <math>\left \lfloor d_2 \right \rfloor = 11</math>, <math>\left \lfloor d_3 \right \rfloor = 1361</math>, ... {{OEIS|id=A051254}}. Very little is known about the constant <math>A</math> (not even whether it is [[rational number|rational]]). This formula has no practical value, because there is no known way of calculating the constant without finding primes in the first place.

There is nothing special about the [[floor function]] in the formula. Tóth proved that there also exists a constant <math>B</math> such that

:<math> \lceil B^{r^{n}} \rceil</math>

is also prime-representing for <math>r>2.106\ldots</math>.<ref>{{citation|first=László|last=Tóth|title=A Variation on Mills-Like Prime-Representing Functions|journal=Journal of Integer Sequences|volume=20|year=2017|issue=17.9.8|arxiv=1801.08014|url=https://cs.uwaterloo.ca/journals/JIS/VOL20/Toth2/toth32.pdf}}.</ref>

In the case <math>r=3</math>, the value of the constant <math>B</math> begins with 1.24055470525201424067... The first few primes generated are:

:<math>2, 7, 337, 38272739, 56062005704198360319209, </math>
:<math> 176199995814327287356671209104585864397055039072110696028654438846269, \ldots</math>

''Without'' assuming the Riemann hypothesis, Elsholtz developed several prime-representing [[function (mathematics)|functions]] similar to those of Mills. For example, if <math>A = 1.00536773279814724017 \ldots</math>, then <math>\left\lfloor A^{10^{10n}} \right\rfloor</math> is prime for all positive integers <math>n</math>. Similarly, if <math>A = 3.8249998073439146171615551375 \ldots</math>, then <math>\left\lfloor A^{3^{13n}} \right\rfloor</math> is prime for all positive integers <math>n</math>.<ref name="Elsholtz">{{citation
 | doi = 10.1080/00029890.2020.1751560
 | first = Christian | last = Elsholtz
 | title = Unconditional Prime-Representing Functions, Following Mills
 | journal = American Mathematical Monthly
 | volume = 127
 | issue = 7
 | pages = 639–642
 | publisher = [[Mathematical Association of America]]
 | location = Washington, DC
 | year = 2020
 | arxiv = 2004.01285| s2cid = 214795216 }}
</ref>