Experimental Number Theory, Part I : Tower Arithmetic

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by  Edinah K. Gnang

Experimental Number Theory
1 Part I : Tower Arithmetic
1
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2 by Edinah K. Gnang

n
a January 18, 2011
J

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1 Introduction

]
T
We introduce in this section an Algebraic and Combinatorial approach to the
N
theoryofNumbers. Theapproachrestsontheobservationthatnumberscanbe
.
h identifiedwithfamiliarcombinatorialobjectsnamelyrootedtrees,whichweshall
t hererefertoastowers. Thebijectionbetweennumbersandtowersprovidessome
a
m insightsintounexpectedconnexionsbetweenNumbertheory,combinatoricsand
discrete probability theory.
[

1
Definition 1.1
v
6 LetXXX denoteandimensionalvectorwhoseentriesaredistinctvariablesdefined
2
by
0
3 XXX =(xk)1≤k≤n , (1)
.
1 a tower expansion ( or simply a tower ) over XXX is a finite product of iterated
0 exponentiations over the entries of XXX. Furthermore the set of towers over the
1
entries ofXXX is denoted (XXX).
1 T
:
v
Example 1.1
i
X
LetXXX =(x,y), the expressions bellow feature three towers
r
a
x, x(yx), x(yx). x(xy) .y(xy). y(yy) . (2)
(cid:16) (cid:17) (cid:16) (cid:17)
The height of a tower1 indicates the maximum number of iterated exponenti-
ations occurring in the tower, the base of the tower refers to the bottom level
of the tower. Finally we will call the variables appearing at the base level of a
tower the pillars of the tower.
1Ishalloftenomittheparenthesisindicatingtheorderinwhichtheiteratedexponentiations
arebeingcarriedout,butweshallalwaysassumethattheiterations areperformedfromthe
topdown.
1
TTThhheeeooorrreeemmm::: FFFuuunnndddaaammmeeennntttaaalllTTThhheeeooorrreeemmmooofffAAArrriiittthhhmmmeeetttiiiccc (((FFF...TTT...AAA...))):::Everypositivein-
teger greater than 1 can be written uniquely as a product of powers of primes.
( the expression written in non decreasing order of the primes.)
CCCooorrrooollllllaaarrryyy ::: Every positive integer greater than 1 can be written uniquely as
a tower expansion over the primes ( the primes at each level of the tower are
written in increasing order ).
we will not discuss here the proof of the F. T. A. but refer the reader to a
beautiful discussion on the proof of the F. T. A. in [1].
Definition 1.2
A Formal Tower Series is a series expansion which consists of a linear com-
bination of distinct but not necessarily finitely many towers. The coefficients
in the linear combination are assumed to originate from a field noted here F (
preferably finite ). The set of such Formal Tower Series is denoted F[ (XXX)].
T
Furthermorea linearcombinationof finitely many distinct towerswill be called
a polytower.
1.1 Revisiting Euler’s product formula for the Riemann
zeta function.
LetXXX denote an infinite dimensional vector of variables defined by
XXX =(x ) (3)
k 1≤k≤∞
AsmentionedearliertheF.T.A.inducesabijectionbetweenN 0,1 andtow-
\{ }
ers over the vector of primes, that bijection in turn suggestsa natural bijection
between N 0,1 and (XXX) as illustrated bellow
\{ } T
T (2) = x
XXX 1
T (3) = x
XXX 2
T (4) = xx1
XXX 1
T (5) = x
XXX 3 (4)
T (6) = x x
XXX 1 2
T (7) = x
XXX 4
.
.
.
Let us now introduce the binary operatorRRR(., .) which we will refer to as the
raiser operator for reason that will be apparent subsequently.
RRR: T (k) F[ (XXX)]֌F[ (XXX)]
{ XXX }1≤k≤∞× T T
RRR x, akTXXX(k) = akxTXXX(k). (5)
 
k∈NX/{0,1} k∈NX\{0,1}
 
2
More generally we write
RRR: T (k) F[ (XXX)]֌F[ (XXX)]
{ XXX }1≤k≤∞× T T
RRR T (l), a T (k) = a (T (l))Tk(XXX). (6)
XXX k XXX k XXX
 
k∈NX/{0,1} k∈NX\{0,1}
 
Formostofourdiscussionhoweverwewillrequirethefirstofthetwodefinition
of raiser operator but we point out that the first definition follows from the
more general following definition.
We recall Euler’s product identity for the Riemann Zeta function as expressed
by
1 p −s −1 =1+ n−s. (7)
k
−
k∈YN\{0}(cid:0) (cid:1) n∈NX\{0,1}
Wewillcometothinkoftheidentityaboveasexpressinganinvarianceprinciple.
One important reasonfor introducing formaltower series is to validate in some
sense identities of the form
pt =1+ n (8)
k
!
k∈N\{0} t∈N n∈N\{0,1}
Y X X
Hereis howthe expressionabovecanbe thoughtto be notonly meaningfulbut
also depicting a fundamental invariance principle.
1+RRR x , 1+ T (n) =1+ T (n) (9)
k XXX XXX
  
k∈YN\{0} n∈NX\{0,1} n∈NX\{0,1}
  
sothattheFormalTowerSeries 1+ T (n) isinvariantunderthe
n∈N\{0,1} XXX
action of the operator (cid:16) (cid:17)
P
(1+RRR(x , (cid:5))) (10)
k
k∈YN\{0}
To get a sense of how such an invariance principle could naturally arises we
consider the function.
g(x)=1+x+xx+xxx +xxxx + (11)
···
anduseittoinduceasequenceoffunctionsonthevectorXXX who’sinitialelement
is
G (XXX)= (1+g(x )) (12)
0 k
1≤k≤Ydim{XXX}
and the other elements of the sequence are defined by the following recursion
G (XXX)= (1+RRR(x ,G (XXX))) (13)
n+1 k n
 
1≤k≤Ydim{XXX}
 
3
So that the fundamental invariance principle is re-casted as
lim G (XXX) =1+ T (n) (14)
n XXX
n→∞{ }
n∈NX\{0,1}
The invariance principle follows from the F. T. A where XXX will represent the
2
vector whose entries are the distinct primes arrangedin increasing order .
Algorithms suchas Buchberger’salgorithmincommutative algebraemphasizes
the importance of totally ordering monomials. In our discussion we shall use
theintegerorderingtoinduceanaturalorderingonthetowers. Oncethetowers
are totally ordered it becomes rather straight forward to discuss Tower Arith-
metic. Let us encapsulate the ordering of towers into a metric function d((cid:5),(cid:5))
so as to embed towers into a metric space ( (XXX), d((cid:5),(cid:5))). The metric space
( (XXX), d((cid:5),(cid:5))) allows us express addition ofTtowers through the following rela-
T
tion
for T (p) T (m)
XXX XXX
≥
d(T (m), T (p))=d(0, T (n)) T (m)+T (n)=T (p). (15)
XXX XXX XXX XXX XXX XXX
⇔
and we use the convention
T (0)=1 and T (0)+T (0)=x (16)
XXX XXX XXX 1
Furthermore multiplication of tower follows immediately from the definition of
addition and it is expressed by
xTXXX(mk) xTXXX(nt) =
 k × t 
1≤k≤Ydim{XXX} 1≤t≤Ydim{XXX}
   
x(TXXX(mk)+TXXX(nk)) (17)
 k 
1≤k≤Ydim{XXX}
 
In summary the base level of the product of tower is the union of the base
level of the towers being multiplied while the powers of corresponding pillars
are added.
Wenowconsiderthecaseoffinitedimensionalvectors. LetPPP beafinitedimen-
sional vector whose entries are made of the smallest dim PPP distinct primes.
{ }
For convenience we arrange the primes in increasing order as entries of PPP we
have
PPP = p , ,p (18)
1 dim{PPP}
···
2the ordering is not necessary for t(cid:0)he invariance princ(cid:1)iple it suffice to have distinct the
entries inXXX ofmustcorresponding todistinctprimes.
4
we define an analogous sequence of functions ofPPP defined by
G (PPP)= (1+g(p )) (19)
0 k
1≤k≤Ydim{PPP}
and the recursion
G (PPP)= (1+RRR(p , G (PPP))) (20)
n+1 k n+1
1≤k≤Ydim{PPP}
in which case we obtain
lim G (PPP) =1+ a T (k) (21)
n k PPP
n→∞{ }
k∈NX/{0,1}
wherea 0,1 ,morespecificallya =1ifthe towerexpansionofn isatower
k k
∈{ }
overPPP and a =0 otherwise.
k
The preceding discussion raises the following interesting question: Consider-
ing a given finite set of consecutive of integers bounded by n. What is the
probability that a number chosen at random contains a particular prime p in
it’s tower expansion
We now propose a theoremwhich follows from Euler’s argumentin his proofof
the Infinity of the primes.
TTThhheeeooorrreeemmm ::: For every finite dimensional vector PPP = p , ,p whose
1 dim{PPP}
···
entries are made up of distinct primes when considering the limit
(cid:0) (cid:1)
lim G (PPP) =1+ a T (n). (22)
n n PPP
n→∞{ }
n∈NX/{0,1}
we have
1+ a (T (n))−1 < (23)
n PPP
  ∞
n∈NX/{0,1}
 
PPPrrroooooofff : The convergence follows immediately from the fact that
1+ a (T (n))−1 < 1 p −1 < (24)
n PPP k
   −  ∞
n∈NX/{0,1} 1≤k≤Ydim{PPP}(cid:0) (cid:1)
   
The preceding theorem suggests that the rational numbers must not be too
far out of our reach once we are equipped with a concrete description of the
5
integers as towers. we recall that for a vector PPP = p , ,p whose
1 dim{PPP}
···
entries are made up of distinct primes. We consider the sequence
(cid:0) (cid:1)
G (PPP)= (1+g(p )) (25)
0 k
1≤k≤Ydim{PPP}
G (PPP)= (1+RRR(p , G (PPP))) (26)
n+1 k n
1≤k≤Ydim{PPP}
This sequence may be used to induce the sequence H defined by
n
H (PPP)= RRR p −1, G (PPP) +1+ RRR(p ,G (PPP)) (27)
n k n k n
1≤k≤Ydim{PPP}(cid:0) (cid:0) (cid:1) (cid:1)
So that the terms in the resulting expression are given by
Lim H (PPP) =1+ a T (q). (28)
n→∞ n q PPP
{ }
q∈QX\{0,1}
If we seek the complete bijection with the rational we would have started with
theinfinite dimensionalvectorXXX insteadandconsideredthe followingsequence
G (XXX)= (1+g(x )) (29)
0 k
1≤k≤Ydim{XXX}
G (XXX)= (1+RRR(p , G (XXX))) (30)
n+1 k n
1≤k≤dim{XXX}
Y
This sequence may be used to induce the sequence H defined by
n
H (XXX)= RRR p −1, G (XXX) +1+ RRR(p , G (XXX)) (31)
n k n k n
1≤k≤Ydim{XXX}(cid:0) (cid:0) (cid:1) (cid:1)
towers in bijections set Q 0,1 are described by
\{ }
Lim H (PPP) =1+ T (q). (32)
n→∞ n PPP
{ }
q∈QX\{0,1}
1.2 Tower Sieve Algorithm
Sieves play an important role in Number theory, We propose to investigate
here a novel sieve algorithm based on the arithmetic of towers. Let PPP =
p , ,p denote vector whose entries are all the primes less the 2t for
1 dim{PPP}
···
some 1 t, the goal is to determine the primes in the range 2t,21+t . The
(cid:0) ≤ (cid:1)
algorithm consists in computing the following recursion
(cid:2) (cid:3)
g (x)=1+x+xx+ + xx.··x last term of height s (33)
s
···
(cid:16) o (cid:17)
6
G (PPP)= g (p ) (34)
0 sk k
1≤k≤Ydim{PPP}
wherepkpk.··pkisthelasttermintheexpressiongsk(pk)issuchthatpkpk.··pk ≪
21+t
G (PPP)= (1+RRR(p , G (PPP))) (35)
n+1 k n
 
1≤k≤Ydim{PPP}
 th 
we stop the recursion at the m iteration if all the towers remaining in the
polytower difference (G (PPP) G (PPP)) are are towers greater than 21+t.
m+1 m
−
Attheheartofthe recursivealgorithmisthe factthattherecursiondetermines
the tower expansions of integers in the interval 2t, 21+t with the exception of
towersexpansionwhichcontainsprimeswhicharenotlessthan2t. Furthermore
(cid:2) (cid:3)
assuming that we order the towers the gaps of size 2 in the list determines the
exact location of primes in the range of interest. This provide a constructive
proof of the fact that there is always at least one prime in the range 2t, 21+t
for any value of 1 t. The sieves algorithm we discussed above is rather dif-
≤ (cid:2) (cid:3)
ferent from Eratosthenes sieve in that it generates the composite and indicates
exactlywhere the primes oughtto be foundandmore importantlyas opposeto
some variants of Eratosthenes sieve methods our algorithm ensures that each
composite is generated exactly once. Let us briefly go through the steps the
algorithm with the case t=2 for illustration purposes
1.3 Illustration of the Algorithm
We illustrate the algorithm using Mathematica.
LetXXX =(x ,x ) the Mathematica commands used are
1 2
ggg :::===111+++xxx
888 kkk
FFFooorrr[[[iii===111,,,iii<<<777,,,iii++++++,,,ggg ===(((111+++TTToootttaaalll[[[(((xxx ∧∧∧LLLiiisssttt@@@@@@(((ggg))))))]]])))]]]
888 kkk
from whichggg is given by
888
ggg =1+x +xxk +xxxkk +xxxkxkk +xxxkxkxkk +xxxkxkxkxkk +xxxkxkxkxkxkk (36)
k k k k k k k
The Mathematica commands for the recursion are given by
GGG :::===111
000
FFFooorrr[[[kkk===111,,,kkk<<<333,,,kkk++++++,,,GGG ===EEExxxpppaaannnddd[[[GGG (((111+++xxx )))]]]]]]
000 000 kkk
after these first commands we have
GGG =1+x +x +x x (37)
000 1 2 1 2
7
Here is an overview of the typical intermediary steps required to compute the
recursion.
LLLiiisssttt@@@@@@(((GGG )))= 1,x ,x ,x x
000 1 2 1 2
{ }
111,,,xxx ,,,xxx ,,,xxx xxx ///...xxx 222///...xxx 333= 1,2,3,6
111 222 111 222 111 222
{{{ }}} →→→ →→→ { }
(((xxx ∧∧∧LLLiiisssttt@@@@@@(((GGG ))))))= x ,xx1,xx2,xx1x2
111 000 { 1 1 1 1 }
TTToootttaaalll[[[(((xxx ∧∧∧LLLiiisssttt@@@@@@(((GGG ))))))]]]=x +xx1 +xx2 +xx1x2
111 000 1 1 1 1
EEExxxpppaaannnddd[[[(((111+++TTToootttaaalll[[[(((xxx ∧∧∧LLLiiisssttt@@@@@@(((GGG ))))))]]])))(((111+++TTToootttaaalll[[[(((xxx ∧∧∧LLLiiisssttt@@@@@@(((111+++xxx ))))))]]])))]]]
111 000 222 111
=1+x +xx1 +xx2 +xx1x2 +x +x x +xx1x +xx2x +
1 1 1 1 2 1 2 1 2 1 2
xx1x2x +xx1 +x xx1 +xx1xx1 +xx2xx1 +xx1x2xx1
1 2 2 1 2 1 2 1 2 1 2
The polytower contains the towers of interest and the list bellow depicts the
corresponding numbers.
LLL000===SSSooorrrttt[[[[[[LLLiiisssttt@@@@@@EEExxxpppaaannnddd[[[(((111+++TTToootttaaalll[[[(((xxx ∧∧∧LLLiiisssttt@@@@@@(((EEExxxpppaaannnddd[[[(((111+++xxx )))(((111+++xxx )))]]]))))))]]])))
111 111 222
(((111+++TTToootttaaalll[[[(((xxx ∧∧∧LLLiiisssttt@@@@@@(((111+++xxx ))))))]]])))]]]///...xxx 222///...xxx 333]]]
222 111 111 222
→→→ →→→
= 1,2,3,4,6,8,9,12,18,24,36,64,72,192,576 (38)
{ }
If we add in the primes determined by the list we get the following sequence of
towers and their corresponding numbers.
FFF:::===
EEExxxpppaaannnddd[[[(((111+++TTToootttaaalll[[[(((xxx ∧∧∧LLLiiisssttt@@@@@@(((EEExxxpppaaannnddd[[[(((111+++xxx )))(((111+++xxx )))]]]))))))]]])))(((111+++TTToootttaaalll[[[(((xxx ∧∧∧LLLiiisssttt@@@@@@(((111+++xxx ))))))]]])))
111 111 222 222 111
(((111+++xxx )))(((111+++xxx )))]]]
333 444
if we list the resulting towers we get
LLLiiisssttt@@@@@@FFF =
1,x ,xx1,xx2,xx1x2,x ,x x ,xx1x ,xx2x ,xx1x2x ,xx1,x xx1,
{ 1 1 1 1 2 1 2 1 2 1 2 1 2 2 1 2
xx1xx1,xx2xx1,xx1x2xx1,x ,x x ,xx1x ,xx2x ,xx1x2x ,x x ,x x x ,
1 2 1 2 1 2 3 1 3 1 3 1 3 1 3 2 3 1 2 3
xx1x x ,xx2x x ,xx1x2x x ,xx1x ,x xx1x ,xx1xx1x ,xx2xx1x ,
1 2 3 1 2 3 1 2 3 2 3 1 2 3 1 2 3 1 2 3
xx1x2xx1x ,x ,x x ,xx1x ,xx2x ,xx1x2x ,x x ,x x x ,xx1x x ,
1 2 3 4 1 4 1 4 1 4 1 4 2 4 1 2 4 1 2 4
xx2x x ,xx1x2x x ,xx1x ,x xx1x ,xx1xx1x ,xx2xx1x ,xx1x2xx1x
1 2 4 1 2 4 2 4 1 2 4 1 2 4 1 2 4 1 2 4
,x x ,x x x ,xx1x x ,xx2x x ,xx1x2x x ,x x x ,x x x x ,
3 4 1 3 4 1 3 4 1 3 4 1 3 4 2 3 4 1 2 3 4
xx1x x x ,xx2x x x ,xx1x2x x x ,xx1x x ,
1 2 3 4 1 2 3 4 1 2 3 4 2 3 4
x xx1x x ,xx1xx1x x ,xx2xx1x x ,xx1x2xx1x x (39)
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4}
The corresponding list of integer is determined by the following Mathematica
commands
8
LLL111===SSSooorrrttt[[[LLLiiisssttt@@@@@@FFF///...xxx 222///...xxx 333///...xxx 555///...xxx 777]]]
111 222 333 444
→→→ →→→ →→→ →→→
which yields the following list of integers.
1,2,3,4,5,6,7,8,9,10,12,14,15,18,20,21,24,28,30,35,
{
36,40,42,45,56,60,63,64,70,72,84,90,105,120,126,140,168,180,
192,210,252,280,315,320,360,420,448,504,576,630,840
,960,1260,1344,2240,2520,2880,4032,6720,20160 (40)
}
We note that the simple insertion of these towers determines the prime
number 11 and 13. The next step is to discuss a slight modification of the
straightforward technique discussed above . We see from the algorithm that
we are very strongly incentives to reduce the number of terms appearing in the
expressionfor instance it is clear that the number shouldnot be included sowe
subs-tract it and consider the following expression .
AAA:::===111+++TTToootttaaalll[[[(((xxx ∧∧∧LLLiiisssttt@@@@@@(((EEExxxpppaaannnddd[[[(((111+++xxx )))(((111+++xxx )))]]] xxx xxx ))))))]]]
111 111 222 111 222
−−−
The previous would correspond to what I would refer to as a renormalization
step
BBB:::===111+++xxx
222
The list of towers and the corresponding list of numbers generated by the re-
duced product is given by performing by re-normalizing as follows
EEExxxpppaaannnddd[[[AAABBB]]] xxxxxx111xxx xxxxxx222xxx
−−− 111 222−−− 111 222
=1+x +xx1 +xx2 +x +x x (41)
1 1 1 2 1 2
SSSooorrrttt[[[LLLiiisssttt@@@@@@(((EEExxxpppaaannnddd[[[AAABBB]]] xxxxxx111xxx xxxxxx222xxx )))///...xxx 222///...xxx 333]]]
−−− 111 222−−− 111 222 111→→→ 222→→→
= 1,2,3,4,6,8 (42)
{ }
Now if we want to improve the estimate of the sum of the primes in the range
and than we must removefrom the sum the towersassociatedwith the number
in the range of 1 and as follows
EEExxxpppaaannnddd[[[(((111+++TTToootttaaalll[[[(((xxx ∧∧∧LLLiiisssttt@@@@@@(((EEExxxpppaaannnddd[[[(((111+++xxx )))(((111+++xxx )))]]]))))))]]] xxxxxx111xxx222)))(((111+++xxx )))]]] xxxxxx111xxx
111 111 222 −−− 111 222 −−− 111 222−−−
xxxxxx222xxx 111 xxx xxx
111 222−−− −−− 111−−− 222
=xx1 +xx2 +x x (43)
1 1 1 2
Here is now the estimate if the sum of the primes in the range , is given by the
following
GGG:::===EEExxxpppaaannnddd[[[AAABBB(((111+++xxx )))(((111+++xxx )))]]]
333 444
LLL222===SSSooorrrttt[[[LLLiiisssttt@@@@@@GGG///...xxx 222///...xxx 333///...xxx 555///...xxx 777]]]
111 222 333 444
→→→ →→→ →→→ →→→
= 1,2,3,4,5,6,7,8,10,12,14,15,20,21,24,28,30,35,40,
{
9
42,56,60,70,84,105,120,140,168,210,280,420,840 (44)
}
We note that the impact of the reduction is significant we have almost reduced
the length we are considering by a factor of 2 as depicted bellow which in of
itself is pretty amazing.
DDDiiimmmeeennnsssiiiooonnnsss[[[LLL111]]]
= 60 (45)
{ }
DDDiiimmmeeennnsssiiiooonnnsss[[[LLL222]]]
32 (46)
{ }
if one just wants to generate the trees without any concern for the the renor-
malization business here is how one proceeds.
GGG :::===111
000
FFFooorrr[[[kkk===111,,,kkk<<<333,,,kkk++++++,,,GGG ===EEExxxpppaaannnddd[[[GGG (((111+++xxx )))]]]]]]
000 000 kkk
GGG :::===111FFFooorrr[[[kkk===111,,,kkk<<<333,,,kkk++++++,,,GGG ===EEExxxpppaaannnddd[[[GGG (((111+++TTToootttaaalll[[[(((xxx ∧∧∧LLLiiisssttt@@@@@@(((GGG ))))))]]])))]]]]]]GGG
111 111 111 kkk 000 111
= 1+x +xx1 +xx2 +xx1x2 +x +x x +xx1x +xx2x +
{ 1 1 1 1 2 1 2 1 2 1 2
xx1x2x +xx1 +x xx1 +xx1xx1 +xx2xx1 +xx1x2xx1 +xx2 +x xx2+
1 2 2 1 2 1 2 1 2 1 2 2 1 2
xx1xx2 +xx2xx2 +xx1x2xx2 +xx1x2 +x xx1x2 +xx1xx1x2 +xx2xx1x2 +xx1x2xx1x2
1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 }
(47)
which results in the following list of integers.
1,2,3,4,6,8,9,12,18,24,27,36,54,64,72,108,192
{
,216,576,729,1458,1728,2916,5832,46656 (48)
}
The preceding sequence of numbers allowed us to determine that the primes 5
and 7 are missing from the list.
1.4 Recovering the Ordering.
Whatwewouldpresumablylikewouldbetorecovertheorderingoftheintegers
by simply manipulating the tower expansion. This would also mean that we
would recover the values of the primes. this would not be an easy task but it
has the merit of showing how in a sense all the primes are determined by the
first two integers. We therefore define
g (x )=1+x +x xi,t+ + x xi,t.··xi,t last term of height s (49)
si i,t i,t i,t ··· i,t i
(cid:16) o (cid:17)
G (XXX)= (1+g (x )) (50)
0 si i,1
1≤i≤Ydim{XXX}
10