On existence of infinite primes and infinite twin primes

0.08 MB·English

by  Maurice Margenstern

ON EXISTENCE OF INFINITE PRIMES AND
INFINITE TWIN PRIMES
5 Maurice Margenstern andYaroslavD. Sergeyev†‡
∗
1
0
2

b
e Abstract
F
Thetwinprimesconjectureisaveryoldproblem. Tacitlyitissupposed
3
thattheprimesitdealswitharefinite. Inthepresentpaperweconsiderthree
2
problemsthatare notrelatedto finiteprimesbutdealwith infiniteintegers.

] Themaintoolofourinvestigationisanumeralsystemproposedrecentlythat
M
allowsonetoexpressvariousinfinitiesandinfinitesimalseasilyandbyafi-
G nitenumberofsymbols.Theproblemsunderconsiderationarethefollowing
. andforallofthemwegiveaffirmativeanswers: (i)doinfiniteprimesexist?
h
(ii)doinfinitetwinprimesexist?(ii)isthesetofinfinitetwinprimesinfinite?
t
a Examplesofthesethreekindsofobjectsaregiven.
m
[ KeyWords: Infiniteprimes,infinitetwinprimes,infinitesets.

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v
1 Introduction
1
5
0 The research on twin primes number has given rise to an important number of
3
works(see, forexample, [6]andreferences giventherein). Inthispaper, westudy
0
. the problem of the existence of infinite primes and infinite twin primes. So, our
1
0 paper brings in no new light on the traditional twin primes conjecture. Let us
5 mention a few works which can be considered as precursors in some sense and
1
might be thought of an exotic character by pure mathematicians. As an example
:
v of such an exotic issue, we can quote the existence of other natural families of
i
X numbers whose distribution is alike that of primes. There is an example of such
r a family for which an analogue of the twin primes can be formulated and was
a
indeed proved, see[14]. Now,whatwecanconsider asprecursors forusaremore
connected with logical problems. The first paper in this direction is [11], where
Maurice Margenstern, Ph.D., is Professor Emeritus at the Universite´ de Lorraine, e-mail:
∗
[email protected]
†Yaroslav D. Sergeyev, Ph.D., D.Sc., D.H.C., is Distinguished Professor at the University
of Calabria, Rende, Italy. He is also Professor (a part-time contract) at the Lobatchevsky
State University, Nizhni Novgorod, Russia and Affiliated Researcher at the Institute of High
Performance Computing and Networking of the National Research Council of Italy, e-mail:
[email protected]
‡Correspondingauthor.
1
thepossibility ofinfiniteprimenumbersoftheformweconsider inthispaperwas
investigated without reaching definite results. The second paper is [19] where the
author constructs a model of a theory in which it is possible to prove the twin
prime conjecture. There is an important difference in this paper with our result
in this sense that the theory considered by this author assumes a weak induction
axiomwhilehere, wehavenoinduction atalloninfiniteintegers. Theformofthe
twinprimesinthatpaperhassomesimilarity withthatofourinfinitetwinprimes,
although thepaperappealstohigherresultsinalgebraandanalysis whichisnotat
allthecaseofourwork.
Anumeralsystemintroducedrecentlyin[20,21]forperformingcomputations
withinfinities andinfinitesimals isusedheretostudytheproblem oftheexistence
ofinfiniteprimesandinfinitetwinprimes. Itshouldbementionedthatthiscompu-
tationalmethodologyisnotrelatedtonon-standard analysisofRobinsonandhasa
strong applied character. In fact, the Infinity Computer working numerically with
a variety of infinite and infinitesimal numbers has been introduced (see the patent
[23]).
In order to see the place of the new approach in the historical panorama of
ideasdealing withinfiniteandinfinitesimal, see[10,12,13,15,18,22,24,32]. In
particular, connections of the new approach with bijections is studied in [15] and
metamathematical investigations on the theory can be found in [13]. Among the
applications where the new approach has been successfully used we can mention
thefollowing: percolation andbiological processes (see[7,8,34,28]),hyperbolic
geometry (see [16, 17]), numerical differentiation and optimization (see [1, 26,
36]),infiniteseries(see[9,22,27,35]),thefirstHilbertproblem,Turingmachines,
and lexicographic ordering (see[24, 31, 32,33]), cellular automata (see [2,3,4]),
ordinary differential equations (see[30]),etc.
The rest of the paper has the following structure. The next Section contains a
briefinformal description ofthenumeral system allowing onetoexpress different
infinities and infinitesimals in a unique framework (see [21, 25, 29] for a detailed
discussion). Section3presentsmainresultsofthepaper.
2 Infinitiesandinfinitesimalsexpressedingrossone-based
numerals
Letus consider a study published in Science (see [5])where there isa description
of a primitive tribe living in Amazonia - Piraha˜ - that uses a very simple numeral
system1 forcounting: one,two,many.
ForPiraha˜,allquantitieslargerthantwoarejust‘many’andsuchoperationsas
1 We remind that numeral is a symbol or a group of symbols that represents a number. The
differencebetweennumeralsandnumbersisthesameasthedifferencebetweenwordsandthethings
theyreferto.Anumberisaconceptthatanumeralexpresses. Thesamenumbercanberepresented
bydifferent numerals. Forexample, thesymbols ‘3’, ‘three’, and‘III’aredifferent numerals, but
theyallrepresentthesamenumber.
2
2+2 and 2+1 give the same result, i.e., ‘many’. Using their weak numeral system
Piraha˜ are not able to see numbers 3, 4, etc., to execute arithmetical operations
with them, and, in general, to say anything about these numbers because in their
language thereareneither wordsnorconcepts forthat. Moreover, theweaknessof
theirnumeralsystemleadstosuchresultsas
‘many’+1=‘many’, ‘many’+2=‘many’,
which are very familiar to us in the context of views on infinity used in the tradi-
tionalcalculus
¥ +1=¥ , ¥ +2=¥ .
This analogy advices that our difficulty in working with infinity is not connected
to the nature of infinity but is just a result of inadequate numeral systems used to
express numbers. In fact, numeral systems strongly influence our capabilities to
describe physical andmathematical objects. Forinstance, Romannumeral system
has no numeral to express 0. As a consequence, the expression III-X in this nu-
meralsystemisanindeterminateform. Moreover,anyassertionregardingnegative
numbers and zero cannot be formulated using Roman numerals because there are
nosymbolscorresponding totheseconcepts inthisconcretenumeralsystem.
The numeral system proposed in [21, 25, 29] is based on an infinite unit of
measureexpressedbythenumeral① calledgrossoneandintroducedasthenumber
of elements of the set N of natural numbers (a clear difference with non-standard
analysiscanbeseenimmediatelysincenon-standard infinitenumbersarenotcon-
nected to concrete infinite sets and do not belong to N). Other symbols dealing
with infinities and infinitesimals (¥ , Cantor’s w , (cid:192) ,(cid:192) ,..., etc.) are not used to-
0 1
gether with ① . Similarly, when the positional numeral system and the numeral 0
expressing zero had been introduced, symbols V, X, and other symbols from the
Romannumeralsystemhadnotbeeninvolved.
Notice that people very often do not pay a great attention to the distinction
betweennumbers andnumerals (inthisoccasion itisnecessary torecallconstruc-
tivistswhostudiedthisissue),manytheoriesdealingwithinfiniteandinfinitesimal
quantities have asymbolic (notnumerical) character. Forinstance, manyversions
of non-standard analysis are symbolic, since they have no numeral systems to ex-
presstheirnumbers byafinitenumberofsymbols(thefinitenessofthenumberof
symbols isnecessary fororganizing numerical computations). Namely,ifwecon-
siderafinitenthanitcanbetaken n=7,orn=108oranyothernumeral usedto
express finite quantities and consisting of afinite number of symbols. Incontrast,
ifweconsider anon-standard infinitemthen itisnotclear whichnumerals canbe
usedtoassignaconcretevaluetom. Oneoftheimportantdifferences betweenthe
new approach and non-standard analysis consists of the fact that the new numeral
system allows us to assign concrete values to infinities (and infinitesimals) as it
happens with finite values. In fact, we can assign m=① , m=3① 2 or to use
−
any other infinite numeral involving grossone to give a numerical value to m (see
[21,25,29]foradetaileddiscussion).
3
The numeral ① allows one to construct different numerals expressing differ-
ent infinities and infinitesimals and to execute numerical computations with all of
them. Asaresult,inoccasionsrequiringinfinitiesandinfinitesimalsindeterminate
formsandvariouskindofdivergencearenotpresentwhenoneworkswithany(fi-
nite,infinite, orinfinitesimal) numbersexpressible inthenewnumeralsystemand
it becomes possible to execute arithmetical operations with a variety of different
infinities and infinitesimals. For example, for ① and ① 3.1 (that are examples of
infinities)and① 1 and① 3.1 (thatareexamplesofinfinitesimals) itfollows
− −
①
0 ① =① 0=0, ① ① =0, =1, ① 0=1, 1① =1, 0① =0, (1)
· · − ①
0 ① 1=① 1 0=0, ① 3.1>① 1>1>① 1>① 3.1>0,
− − − −
· ·
① 1 5+① 3.1
① 1 ① 1=0, − =1, − =5① 3.1+1, (① 1)0=1,
− − − ① 1 ① 3.1 −
− −
① 3.1+4①
① ① 1=1, ① ① 3.1=① 2.1, =① 2.1+4,
− − − ①
· ·
① 3.1
=① 6.2, (① 3.1)0=1, ① 3.1 ① 1=① 2.1, ① 3.1 ① 3.1=1.
① 3.1 · − · −
−
Itfollowsfrom(1)that① 0=1,therefore,afinitenumberacanberepresented
in the new numeral system simply as a① 0 =a, where the numeral a itself can be
written down by any convenient numeral system used to express finite numbers.
The simplest infinitesimal numbers are represented by numerals having only neg-
ativefinitepowersof① (e.g.,50.1① 10.2+16.38① 20.3,seealsoexamplesabove).
− −
Notice that all infinitesimals are not equal to zero. In particular, ①1 >0 because
it is a result of division of two positive numbers. We shall not speak more about
infinitesimals heresincetheyarenotusedinthepresentpaper.
It should be mentioned that in certain cases ① -based numerals allow us to ex-
ecute a finer analysis of infinite objects than traditional tools allow us to do. For
instance, it becomes possible to measure certain infinite sets and to see, e.g., that
the sets of even and odd numbers have ① /2 elements each. The set Z of integers
has2① +1elements(① positiveelements,① negativeelements,andzero). Within
thecountablesetsandsetshavingcardinality ofthecontinuum (see[12,24,25])it
becomes possible to distinguish infinite sets having different number of elements
expressible inthenumeralsystemusinggrossone andtoseethat,forinstance,
①
<① 1<① <① +1<2① +1<2① 2 1<2① 2<2① 2+1<
2 − −
① ① ① ① ① ① ①
2① 2+2<2 1<2 <2 +1<10 <① 1<① <① +1.
− −
Itisimportanttostress thatthenewapproach doesnotcontradict Cantor’s results.
The situation is similar to what happens when one uses a microscope with two
different lenses: the first of them is weak and allows one to see the object of the
4
observation as two dots and another lens is stronger and allows the observer to
see instead of the first dot 10 dots and instead of the second dot 32 dots. Both
lenses give two correct answers having different accuracies. Analogously, both
approaches, Cantor’s and the new one, give correct answers but the accuracy of
the answers is different. Cantor’s tools say that the sets of even, odd, natural, and
integer numbers have the same cardinality (cid:192) . This answer is correct with the
0
precisionthatcardinalnumbershave. However,thefactthattheyallhavethesame
cardinality canbeviewedalsoastheaccuracy oftheusedinstrument istoolowto
see that these sets have different numbers of elements. The new numeral system
allowsustoseethesedifferencesamongsetshavingcardinality(cid:192) andamongsets
0
havingcardinality ofthecontinuum, aswell(see[12,15,24,25,29]foradetailed
discussion including alsotheone-to-one correspondence issues).
We conclude this brief informal introduction by mentioning that properties of
grossonearedescribedbytheInfiniteUnitAxiom(see[24,25,29])thatisaddedto
axiomsforrealnumbers. Inthecontext ofthepresent paper twoissues postulated
bytheAxiomareimportantforus: grossone isaninfinitenumber;(ii)grossone is
divisiblebyanyfiniteinteger. Noticethatgrossoneisnottheonlynumberenjoying
thelatterproperty. Infact,zeroisalsodivisible byanyfiniteinteger.
3 Infinite primes and twin primes
We need some definitions and conventions to continue our study. First, in the
further consideration we use the following representation of an infinite number c
whereitsinfinitepartisseparatedfromitsfinitepart: c=c +c . Inthisseparation,
1 2
c isinfiniteandisexpressedbynumeralsinvolving① andc isfiniteandisrepre-
1 2
sented bynumerals used towrite downfinitenumbers. Note that such arepresen-
tation isnot unique. Forinstance, the number k=1.7① 1.5can bedecomposed
as k =1.7① , k = 1.5, or as k˜ =1.7① 1, k˜ = 0−.5, or in some other way.
1 2 1 2
− − −
Clearly, thisdecomposition becomes unique ifwerequire thatthepartc doesnot
1
containanypartexpressedbyfinitenumeralsonly. Inourexamplewiththedecom-
positionofthenumberkwehaveitsuniquedecomposition k =1.7① , k = 1.5.
1 2
−
Then,infinitenumbersthatdonotcontainfinitepartsarecalledpurelyinfinite.
In other words, this means that in their unique decomposition they have c = 0.
2
Infinite numbers having their infinite part c including more than one infinite part
1
represented by different powers of ① are called compound. Numbers that are not
compound are called simple. For instance, the numbers ① 3① 12 and ① 2+① +
−
3.5 are both compound; the former is purely infinite while the latter is not. The
①
numbers and① 2+1areexamplesofsimpleinfinitenumbers; againtheformer
2
ispurelyinfinitewhilethelatterisnot. Finally,ifafiniteorinfinitenumbercisthe
squareofanintegerd,i.e.,c=d2,wesayhereinafter simplythatcisasquare.
Lemma 3.1. There exist purely infinite simple numbers l divisible by all finite
integers.
5
Proof. Duetoitsdefinition① issuchanumberl . Then,foranypositivefinite
①
numbern, isalsoanexampleofsuchanumberl . Infact,foranyfinitenumber p
n
①
theproduct pnisfiniteand,therefore,itfollows pn① and,asaconsequence, p .
| | n
① 2
Analogously, ① 2 and foranypositivefinitenumbernareexamplesofl . ✷
n
Theorem3.1. Forallpurelyinfinite simplepositive integers l suchthatanyfinite
positiveinteger dividesl itfollowsthatl +1isaprimenumber.
Proof. Let us consider the number l +1, where l is a purely infinite simple
positiveintegersuch pdividesl ,whateverthefinitenumber pis. Letusshowthat
therearenointegersaandbsuchthata b=l +1.
·
Suppose that such integers exist. Then two situations are possible: (i) a is
finiteandbisinfinite; (ii)both aand bareinfinite. Thefirstsituation cannot hold
becausel isdivisiblebyanyfinitenumber p. Thereforeif pl +1,asalso pl then
| |
p1whichisimpossible. Andso,l +1cannothaveanontrivialfinitedivisor.
|
Thisfacthasitsimportancealsoforthecase(ii)whereaandbarebothinfinite.
Itmeans they they cannot have finite divisors. Suppose the opposite, i.e., a=k c
·
where k is a finite integer and c is an infinite integer. Then l 1=k cb, i.e., we
− ·
havethatl 1isproductofthefiniteintegerkandtheinfiniteintegercb. Sincekis
−
finite,duetoassumptions ofthelemmak l and,therefore, k l 1. Theobtained
| 6| −
contradiction provesthatinfinitenumbersaandbcannothavefinitedivisors.
Letuslook atthe case (ii)and consider the unique decomposition of numbers
aandbintheform
a=a +a , b=b +b , (2)
1 2 1 2
wherethefirstpartsarepurelyinfiniteintegers (simpleorcompound) andthesec-
ondpartsarefiniteintegers. Thenitshouldbe
a b=(a +a )(b +b )=l +1,
1 2 1 2
·
a b +a b +b a +a b =l +1. (3)
1 1 2 1 2 1 2 2
Letusdenotetheleft-hand partof(3)byLandtheright-hand partof(3)byR.
Note that a b > a b and that a b > b a as a and b are finite integers.
1 1 2 1 1 1 2 1 2 2
| | | | | | | |
Alsonotethata b =0isimpossibleasLwouldcontainnofinitepartwhileRdoes.
2 2
a b a b
1 1 1 1
Consequently, | | and | | are both infinite numbers. This means that in L,
a b b a
2 1 2 1
| | | |
thethree termsa b ,a b andb a areinfinite numbers whereoneofthem, a b ,
1 1 2 1 2 1 1 1
isofahigherorderthantheothersandtheydonotcontainafinitepartinthesense
of the unique decomposition of 2. Consequently, a b = 1 and a b +b a =0
2 2 2 1 2 1
since a b +b a is a smaller infinite than a b and l is a purely infinite simple
2 1 2 1 1 1
number. Asa and b areintegers, wegeta =b =1ora =b = 1. Possibly
2 2 2 2 2 2
−
changing the sign of a and b wemay assume that a =b =1. Thisentails that
1 1 2 2
a +b =0whichgivesus a2=l whichisimpossible asl ispositive. ✷
1 1 − 1
6
① ① 2
Corollary 3.1. For all finite positive integers n, +1 and +1 are infinite
n n
primenumbers.
Thus, we have proved the existence of infinite prime numbers and have given
examples ofsuch numbers expressible in① -based numerals. Letus consider now
theproblemoftheexistenceofinfinitetwinprimes.
Lemma 3.2. For all purely infinite simple positive integers l such that any finite
positive number divides l the infinite integer l 1 is a prime number if and only
−
ifl isnotthesquareofaninteger.
Proof. Let us repeat the argument of Theorem 3.1 and find the factors of the
numberl 1. Namely,wehavethatab=l 1,a=a +a ,b=b +b andasin
1 2 1 2
− −
the proof of Theorem 3.1 we obtain that a b = 1and, therefore, it follows that
2 2
−
a =b anda2=l . Ifl isnotasquare,thisisimpossibleandso,wegetthatl 1
1 1 1 −
isaprimenumber. Now,ifl isasquare,weobtainl =(a 1)(a +1)wherethe
1 1
−
integera satisfiesa =√l and,therefore, l 1cannotbeaprimenumber. ✷
1 1
−
Let us give an example. The infinite integer ① 2 can be taken as the number l
and it can be easily seen that ① 2 1 is not prime since ① 2 1=(① 1)(① +1)
− − −
where① 1and① +1areinfiniteintegers.
−
Lemma 3.3. For all purely infinite simple positive integers l such that any finite
positivenumberdividesl anditisasquareitfollowsthattheinfinitenumber l
p2m+1
cannotbeasquareforfinitemand pwhere pisaprimenumber.
Proof. Supposethat l isasquare. Thenitfollows l =r2,where,since
p2m+1 p2m+1
p2m+1 isafinitenumber, risaninfiniteinteger. Thus,wecanwrite
l = p2m+1 r2=(pmr)2 p.
· ·
Since pis afinite prime number, it cannot be asquare. This result contradicts the
fact that l is the square of an infinite integer and, therefore, we have proved that
l cannotbeasquare. ✷
p2m+1
l
Obviously, theinfinitenumber isanexampleillustrating Lemma3.3.
22m+1
Theorem3.2. Forallpurelyinfinite simplepositive integers l suchthatanyfinite
positive number divides l and l is a square it follows that the numbers l 1
p2m+1 −
l
and +1areinfinitetwinprimesforallpositivefinitenumbersmand pwhere
p2m+1
pisaprimenumber.
Proof. By repeating the argument of Theorem 3.1 we find that the number
l l
+1isprime. ItfollowsfromLemma3.3that isnotasquare. Thus,due
p2m+1 p2m+1
toLemma3.2itfollowsthat l 1isprime. ✷
p2m+1 −
① 2 ① 2
Numbers 1 and +1 are examples of infinite twin primes for all
22m+1 − 22m+1
positivefinitenumbersm.
7
NoticethatinTheorems3.1and3.2weneedonlytoassumethatl isdivisible
byanyfiniteprimenumber. Inthefollowingtheoremtheassumptionthatanyfinite
positiveintegerdividesl becomesessential.
Theorem3.3. If:
(i) l is an infinite simple positive integer such that any finite positive integer di-
videsl ;
(ii)l isasquare;
(iii)misapositive integer;
(iv) pisafiniteprimenumber;
then:
(i)thesetsA(p)haveinfinitely manyelements,where
l
A(p)= x:x= , p2m+1 l ; (4)
{ p2m+1 | }
(ii)allnumbersx A(p)areinfiniteintegers;
∈
(iii)thenumberofelementsofthesetA(p)is
M(p)=max m:y p2m+1=l , (5)
{ · }
where p yand p2∤ y.
|
Proof. SupposethatthesetA=A(p)hasM=M(p)elements andM isfinite.
Since numbers p2m+1 are strictly increasing, p2M+1 is the largest element in the
set. Letusconsiderthenumber p2M+2. SinceMisfinite,itfollowsthatif p2M+2 l ,
|
it should belong to A. However, M+1 > M, thus it cannot belong to A. This
contradiction concludes theproofofthefirstassertionofthetheorem.
Letusprovenowthatalltheelements ofthesetAareinfiniteintegers. Ifmin
l
(4)is a finite integer then the respective number y= is obviously an infinite
p2m+1
integer. Supposenowthatmisinfiniteandyisfinite. Remindthatl isdivisibleby
l
allfinitenumbers. Thus, should bedivisible byallfinitenumbersexcluding,
p2m+1
probably, p. This means that y cannot be finite since in this case it would be
divisible onlybyafinitenumberofintegers.
Letusprovethe thirdassertion ofthe theorem. Thefact p yfollows from our
|
supposition thatl isasquare. Supposenowthaty= p2y . Thenweobtainthat
1
y p2 p2M+1=y p2(M+1)+1=l .
1 1
· · ·
Thiscontradict thefactthatM isthemaximalnumbersuchthat p2M+1 l . ✷
|
Corollary3.2. ResultsofLemma3.3andTheorem3.2holdforinfinitevaluesofm.
8
Corollary 3.3. Thesets
① 2
B(p)= x 1, x+1:x= , p2m+1 ① 2 , (6)
{ − p2m+1 | }
ofinfinite primenumbers have2M(p)elements wheretheinfinite numberM(p)is
from(5).
Forinstance, itfollowsthattheset
① 2
B(2)= x 1, x+1:x= , 22m+1 ① 2 ,
{ − 22m+1 | }
consists ofinfiniteprimenumbersandhasinfinitelymanyelements.
Weconclude the paper with the following rather obvious remark: substituting
① 2 in (6) by ① 4,① 16 or by any other infinite simple positive integer l being a
squarewecangenerateotherinfinitesetsofinfiniteprimenumbers.
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