ABSTRACT (original article): Let
∞
(
c
k
)
k=1
be a sequence of integers. In this note, We investigate the set of
A>1
such that the integer part of
c
1
...
c
k
A
is always a prime number for every positive integer
k
. Let
(
c
k
)
be this set. The first goal of this article is to determine the topological structure of
(
c
k
)
. Under some conditions on
∞
(
c
k
)
k=1
, we reveal that
(
c
k
)⋂[0,a]
is homeomorphic to the Cantor middle third set for some
a
. The second goal is to propose an algebraically independent subset of
(
c
k
)
if
c
k
is rapidly increasing. As a corollary, we disclose that the minimum of
(k)
is transcendental. In addition, we apply the main result to
(
c
k
)
in the case when
c
1
c
2
...
c
k
=
k!
3
. As a consequence, we give an algebraically independent and countably infinite subset of this set. Furthermore, we also get results on the rational approximation, -linear independence, and numerical calculations of elements in
(
c
k
)
. CITATION (original article): Saito, K., & Takeda, W. (2022). Topological properties and algebraic independence of sets of prime-representing constants. Mathematika, 68(2), 429–453. DOI: https://doi.org/10.1112/mtk.12142

Introduction

Let
A
be the minimum constant
A>1
such that the integer part of
k!
3
A
is a prime number for every positive integer
k
. Let
B
be the minimum constant
B>1
such that the integer part of
k!
B
is a prime number for every positive integer
k
. In Proposition 8.4 and Proposition 8.5 in [ST22], we computed that
A=1.3052998807...,B=2.2419914653...
Further, in [ST22], we claimed that the decimal expansion of
A
can be computed 86 decimal places and one of
B
can be computed 254 decimal places. Actually, in this note, we will show that
A=
1.3052998807 9491978403 3294841059 1296663686 2549608006 1597017618 5206288070 6139915993 596734 ⋯
B=
2.2419914653 5780139806 2070832363 7966564773 4221686725 1975427342 1719561351 2956330454 8963305242 7921137903 2094019309 2388005521 2594365689 5941934804 8880868235 5795982761 7894902314 1963845231 5592493198 0018254448 2978475167 9281717230 8197395093 7464511722 1703813613 5023...
Remark that we do not assume any hypotheses in the calculation of
A
but we assume the Riemann hypothesis in the calculation of
B
. Let us propose our program to find the decimal places of
A
and
B
.

Algorithm

Let
(c[k])
k∈
be the sequence of positive integers which satisfies that
(1)
c[k]
is monotonically increasing, that is,
c[k]<=c[k+1]
for all
k∈
;
(2)
c[1]>=1
and
c[k+1]>=2
for all
k∈
;
(3)
c[
k
0
]>=3
for some positive integer
k
0
.
The following algorithm is to calculate the minimum
D>1
such that the integer part
c
1
⋯
c
k
D
is always a prime number. This calculation is based on Lemma 8.9 in [ST22]. We omit the theoretical details here. See [ST22] if the readers want to know more.
1. Start with
p[1]=2
; Input:
(c[k]),M
.
2. Find the minimum prime number
c[k+1]
p[k]
<=p<
c[k+1]
(p[k]+1)
-1.
​Let
p[k+1]=p
. If
k+1=M
, then stop. If not, substituting
k=k+1
and do Step 2 again;
3. If
c[M+1]>=1438989
, then substitute
underRH=0
. If not, substitute
underRH=1
;
If
underRH=0
, then we can unconditionally compute
D
.
If
underRH=1
, then we should assume the Riemann hypothesis to compute
D
.
4. Compare the decimal expansions
1/c[1]...c[M]
p[M]
and
1/c[1]...c[M]
(p[M]+1)
;​Output:
p[1],p[2],...,p[M]
, the decimal expansion of
D
, the number of decimal places, and under RH or not. ​Remark that the lower bound for
c[M+1]
in Step 4 comes from [Mat17]. By this article, we see that each interval
[
c
n
,
c
(n+1)
]
contains at least one prime number if
c=1438989
. In Step 5, we can compute
D
by comparing the decimal expansions
1/c[1]...c[M]
p[M]
and
1/c[1]...c[M]
(p[M]+1)
since we have
1/c[1]...c[M]
p[M]
<=D<
1/c[1]...c[M]
(p[M]+1)
.

Do the algorithm for computations of
A
and
B

Program

In[]:=
Needs["PrimalityProving`"];
In[]:=
step1:=(p[1]=2;)
In[]:=
step2:=Fork=1,k<=M-1,k++,​​start=p[k]^c[k+1]+1;end=(p[k]+1)^c[k+1];​​​​Forn=start,n<end,n++,​​ IfProvablePrimeQ[n],​​ p[k+1]=n;n=end​​ ;​​ ;​​;
In[]:=
step3:=(​​If[c[M+1]>=1438989,underRH=0,underRH=1];)
In[]:=
step4:=(​​prod=Product[c[k],{k,1,M}];​​lower=p[M]^(1/prod);upper=(p[M]+1)^(1/prod);​​m=1;stop=0;​​While[stop==0,​​If[RealDigits[lower,10,m]!=RealDigits[upper,10,m],​​stop=1,​​m++​​];​​];)
In[]:=
output:=((*Output*)​​For[k=1,k<=M,k++,Print["p[",k,"] = ",p[k]];];​​Print["D = ",N[lower,m-1]," (",m-2," decimal places)"];​​If[underRH==0,Print["Unconditional."],​​Print["We should assume the Riemann hypothesis."]];)

Computation of
A

In[]:=
(*Input*)c[k_]:=Which[k==1,3,k>1,3^((k-1)*(k-1)!)];M=3;
In[]:=
step1;step2;step3;step4;output
p[1] = 2
p[2] = 11
p[3] = 2253240236044012487937308538033349567966729852481170503814810577345406584190098644951
D = 1.30529988079491978403329484105912966636862549608006159701761852062880706139915993596734 (86 decimal places)
Unconditional.
Therefore we obtain
​
​
A=
1.3052998807 9491978403 3294841059 1296663686 2549608006 1597017618 5206288070 6139915993 596734 ⋯

Computation of
B

In[]:=
(*Input*)c[k_]:=k;M=6;
In[]:=
step1;step2;step3;step4;output
p[1] = 2
p[2] = 5
p[3] = 127
p[4] = 260144663
p[5] = 1191446656198440121121655815832664828439647
p[6] = 2860537309330136924507695910544916692360026206366532315082985879027479927970873807930928393826668483633374776554402847729964053551099849694177176333430961929381843271753903847984988012861491790006482201816772286979169965363873370771862463683452060079229
D = 2.24199146535780139806207083236379665647734221686725197542734217195613512956330454896330524279211379032094019309238800552125943656895941934804888086823557959827617894902314196384523155924931980018254448297847516792817172308197395093746451172217038136135023 (254 decimal places)
We should assume the Riemann hypothesis.
Therefore by assuming the Riemann hypothesis, we obtain:
B=
2.2419914653 5780139806 2070832363 7966564773 4221686725 1975427342 1719561351 2956330454 8963305242 7921137903 2094019309 2388005521 2594365689 5941934804 8880868235 5795982761 7894902314 1963845231 5592493198 0018254448 2978475167 9281717230 8197395093 7464511722 1703813613 5023...

Other computations of prime-representing constants

For all prime numbers of
p
, we define
A
p
as the minimum constant
A>1
satisfying that the integer part of
k!
3
A
is always a prime number, and the integer part of
3
A
is equal to
p
. In the algorithm, if we replace
p[1]
with
p
and do the algorithm, then we obtain
A
p
. For example, let
c[k]
be a sequence of positive integers satisfying
c[1]×⋯×c[k]=
k!
3
and let
p[1]=3
. Then we do the algorithm as follows:
In[]:=
(*Input*)c[k_]:=Which[k==1,3,k>1,3^((k-1)*(k-1)!)];M=3;​​p[1]=3;step2;step3;step4;output;
p[1] = 3
p[2] = 29
p[3] = 28460192274683252067641587184070060152533008953728086009094435531484770427909293744039496370928530635848872578990126737
D = 1.453746447415674656857846061056409778615790472907289282519233806553104728704948217640725882335868580937320951903510719628 (120 decimal places)
Unconditional.
Therefore
​
A
3
=
1.453746447415674656857846061056409778615790472907289282519233806553104728704948217640725882335868580937320951903510719628... (120 decimal places)
Further, setting
p[1]=5,7,11
, we obtain the following:
In[]:=
p[1]=5;step2;step3;step4;output;​​
p[1] = 5
p[2] = 127
p[3] = 255918332559940351320845503267008710842588004889435096713321192227240825879299018572156449790452117701550923928708734743321628589753855196516355043229945488654355366553841
D = 1.712994501731959237558245183016992717759890903086241618588895644177635469022183263291571843406963649600063046104507221160636444092569088537703306464185291197101511046223109 (171 decimal places)
Unconditional.
In[]:=
p[1]=7;step2;step3;step4;output;
p[1] = 7
Therefore,
Note that these constants are algebraically independent [ST22].

Acknowledgment

The first author was supported by JSPS KAKENHI Grant Number JP19J20878.

References

[Mat17] C. Mattner, Prime numbers in short intervals, Australian National University, 2017, available at https://openresearch-repository.anu.edu.au/bitstream/1885/173643/2/Caitlin_Mattner.pdf.
[ST22] K. Saito and W. Takeda, Topological properties and algebraic independence of sets of prime-representing constants, Mathematika 68 (2022), no. 2, 429-453. available at https://doi.org/10.1112/mtk.12142

CITE THIS NOTEBOOK

Computations of prime-representing constants​
by Kota Saito and Wataru Takeda​
Wolfram Community, STAFF PICKS, April 26, 2022
https://community.wolfram.com/t/19965