View in EDS HTML Full Text PDF Full Text

Some Problems and Conjectures in Number Theory

Saved in:
Bibliographic Details
Title: Some Problems and Conjectures in Number Theory
Language: English
Authors: Pathak, H. K.
Source: International Journal of Mathematical Education in Science and Technology. Jan 2008 39(1):77-82.
Availability: Taylor & Francis, Ltd. 325 Chestnut Street Suite 800, Philadelphia, PA 19106. Tel: 800-354-1420; Fax: 215-625-2940; Web site: http://www.tandf.co.uk/journals/default.html
Peer Reviewed: Y
Physical Description: PDF
Page Count: 6
Publication Date: 2008
Document Type: Journal Articles
Reports - Descriptive
Descriptors: Number Concepts, Theories, Mathematical Logic, Validity, Equations (Mathematics)
DOI: 10.1080/00207390701607240
ISSN: 0020-739X
Abstract: In this article, we shall discuss some interesting, viable, meaningful, applicable and productive conjectures and methods to deal with some fundamental results in the theory of numbers.
Abstractor: Author
Number of References: 6
Entry Date: 2007
Accession Number: EJ781065
Database: ERIC
Full text is not displayed to guests.
FullText Links:
  – Type: pdflink
    Url: https://content.ebscohost.com/cds/retrieve?content=AQICAHj0k_4E0hTGH8RJwT4gCJyBsGNe_WN95AvKlDbXJGqwxwFxuItpVgCGRcmtnOMwBJ0WAAAA4jCB3wYJKoZIhvcNAQcGoIHRMIHOAgEAMIHIBgkqhkiG9w0BBwEwHgYJYIZIAWUDBAEuMBEEDGWS_UHFRj4kl1DvbgIBEICBmkluNkr4BopUjX15YDzZ42qEA0BMxPN42U-sauH3Att6ifkwKrmPVvid4BdIyeXxVksZM2ED7oXz4hspTgSl0vvLXnLkEY0jEpdEDJMFOVPdr2NvOEfnp2cNEgH4Y39aMElZhDgL3qPkdlBL74O9ZEKwfCp6ElFG4uuCHLdD3sQMgtD98VipkzcI1i4MsOdc1pkqppmIZ1KkrMI=
Text:
  Availability: 1
  Value: <anid>AN0027777625;imt15jan.08;2019Feb27.14:40;v2.2.500</anid> <title id="AN0027777625-1">Some problems and conjectures in number theory. </title> <sbt id="AN0027777625-2">1. Introduction</sbt> <p>In this article, we shall discuss some interesting, viable, meaningful, applicable and productive conjectures and methods to deal with some fundamental results in the theory of numbers.</p> <p>There exists multitued of conjectures concerning to prime numbers. It is well known that the set of natural numbers is countable. Moreover, every mathematics teacher is well aware of the fact that the set of all primes, being a subset of ℕ (the set of natural numbers), must be countable. But to prove or disprove its countability through a bijection map is, at present, beyond the resources of mathematics. Consequently, there arises a natural question concerning primes [<reflink idref="bib2" id="ref1">2</reflink>]: Is there a simple general formula for the <emph>n</emph>-th prime <emph>p<subs>n</subs></emph> (a formula, that is to say, by which we can calculate the value of <emph>p<subs>n</subs></emph> for any given <emph>n</emph> with less labour than the use of the sieve of Eratosthenes)? This is yet to be answered by the mathematical world. However, we present a highly promising (at least a formulation) conjecture to settle down this long awaited answer in one of our main results. In fact, our affirmative answer to the formulation of <emph>p<subs>n</subs></emph> for any given <emph>n</emph>∈ℕ will help students for making a one-to-one correspondence from ℕ to the set of all prime numbers.</p> <hd id="AN0027777625-3">2. Main results</hd> <p>Recall that a number <emph>a</emph>>1 is called a prime number (or simply a prime) if it has only two positive divisors (namely 1 and <emph>a</emph>). The first few primes are 2, 3, 5, 7, 11. For any ξ>0, let π(ξ) represent the number of primes ≤ ξ. The question as to whether, and to what degree of accuracy, π(ξ) can be approximated by the functions of analysis was well addressed in part 7, chapter 2, section 3 of [<reflink idref="bib5" id="ref2">5</reflink>]. In this book, one can find a very accurate result for π(ξ) (see, for instance, [<reflink idref="bib6" id="ref3">6</reflink>], p. 19]), the methods used being those of complex function theory. It is well known that the number π(ξ) of primes up to ξ has the order of magnitude as ξ→∞.</p> <p>Suppose that the sequence denotes the primes in the increasing order of their magnitude, i.e. etc.</p> <p>We now initiate our discussion with the following:</p> <hd id="AN0027777625-4">Conjecture 1:</hd> <p>Every <emph>n</emph>-th prime <emph>p<subs>n</subs></emph> is a sum of a 3-partition multiple (unique) of 2<sups>0</sups>, 2<sups>1</sups>, 2<sups>2</sups> in some order. To illustrate this conjecture, suppose there is a 3-partition set of <emph>n</emph>, say , the set of all 3-partition sets of <emph>n</emph>, such that</p> <p>Graph</p> <p>In such case, we shall write</p> <p>Graph</p> <hd id="AN0027777625-5">Example 1:</hd> <p>By easy computations one can check that</p> <p>etc.</p> <p>We now introduce the following conjecture without proof. In fact, the well known <emph>Four Square Theorem</emph> due to Lagrange (see, for instance, [<reflink idref="bib3" id="ref4">3</reflink>], p. 375], [<reflink idref="bib6" id="ref5">6</reflink>], p. 145]) which states that 'the diophantine equation is solvable for every <emph>n</emph> ≥ 0' justifies its validity.</p> <p> <bold>Conjecture 2:</bold> Every positive integer is a sum of three nonnegative integers, where two are squares of integers and the third is itself square of an integer or it is a sum of two squares.</p> <hd id="AN0027777625-6">Example 2:</hd> <p>where etc.</p> <hd id="AN0027777625-7">2.1. Characterization of integral powers of some integers</hd> <p>First of all we introduce <emph>n</emph> place forward shift and backward shift operators, denoted by and , respectively, as follows:</p> <p>(an integer) = <emph>n</emph> place forward shifting of the last digit in comparison to the preceding one;</p> <p>(an integer) = <emph>n</emph> place backward shifting of the last digit in comparison to the preceding one.</p> <p>In our subsequent discussion, we shall introduce some iteration involving integral powers of certain integers. Suppose that + © + denotes column wise sum of numerical expressions. We now illustrate the concepts of <emph>forward shift</emph> and <emph>backward shift</emph> operators in the following:</p> <p>(a) To find the sum . First we note that</p> <p>Graph</p> <p>and</p> <p>Graph</p> <p>Thus, we obtain</p> <p>Graph</p> <p>(b) To find . We obtain</p> <p>Graph</p> <p>Let denote . In our further illustrations we simply write <emph>sum</emph> in lieu of . We now state certain iteration analogous to Bernoulli's iteration (see, for instance, [<reflink idref="bib1" id="ref6">1</reflink>], [<reflink idref="bib4" id="ref7">4</reflink>]).</p> <hd id="AN0027777625-8">Conjecture 3:</hd> <p>For each :</p> <p>Graph</p> <hd id="AN0027777625-9">Proof:</hd> <p>The proof follows by induction.</p> <hd id="AN0027777625-10">Example 3:</hd> <p>For ; that is,</p> <p>Graph</p> <p>For ; that is,</p> <p>Graph</p> <p>For ; that is,</p> <p>Graph</p> <p>For ; that is,</p> <p>Graph</p> <hd id="AN0027777625-11">Conjecture 4:</hd> <p>For each :</p> <p>Graph</p> <p>or equivalently</p> <p>Graph</p> <hd id="AN0027777625-12">Example 4:</hd> <p>For ; that is,</p> <p>Graph</p> <hd id="AN0027777625-13">Conjecture 5:</hd> <p>For each :</p> <p>Graph</p> <p>or equivalently</p> <p>Graph</p> <hd id="AN0027777625-14">Proof:</hd> <p>The proof follows by induction.□</p> <hd id="AN0027777625-15">Example 5:</hd> <p>For <emph>n</emph> = 2: ; that is,</p> <p>Graph</p> <hd id="AN0027777625-16">Conjecture 6:</hd> <p>For each :</p> <p>Graph</p> <p>or equivalently</p> <p>Graph</p> <hd id="AN0027777625-17">Proof:</hd> <p>The proof follows by induction.□</p> <hd id="AN0027777625-18">Example 6:</hd> <p>For <emph>n</emph> = 4: ; that is,</p> <p>Graph</p> <hd id="AN0027777625-19">Conjecture 7:</hd> <p>For each :</p> <p>Graph</p> <p>or equivalently</p> <p>Graph</p> <hd id="AN0027777625-20">Proof:</hd> <p>The proof follows by induction.□</p> <hd id="AN0027777625-21">Example 7:</hd> <p>For <emph>n</emph> = 4: ; that is,</p> <p>Graph</p> <hd id="AN0027777625-22">3. Applications</hd> <p></p> <ulist> <item> 1. These conjectures have very important roles to play in binary coding, binomial networking, pattern analysis, cryptography (personal keys), Boolean algebra, BCK-algebra, graph theory, numerical analysis and many other branches of applied sciences, particularly in the study of convex structures, crystallography, spline theory, wavelets theory, computer science and information technology.</item> <p></p> <item> 2. It is well known that bits and bytes play an important role in computer architecture, particularly, in information technology; and the magic of the number 2<sups>10</sups> (=1024) is well known in defining the units kilobyte (KB), megabyte (MB), gigabyte (GB) etc. in terms of bytes, kilobytes, megabytes, respectively.</item> <p></p> <item> 3. The result mentioned in Conjecture 3 is very crucial in binary system. In fact, by suitable programming in software technology the Conjecture 3 will improve computational technique beyond imagination.</item> </ulist> <hd id="AN0027777625-23">Acknowledgements</hd> <p>A part of this work was done while the author was a Visiting Fellow at the University of Transkei, Umtata, S.Africa. The author thanks Prof. A.H. Dye, Dean, Faculty of Research and Prof. S.N. Mishra, Director, School of Mathematical Sciences, for providing the excellent research facilities. The author would also like to thank the referee for his valuable comments and suggestions in improving the article.</p> <ref id="AN0027777625-24"> <title> References </title> <blist> <bibl id="bib1" idref="ref6" type="bt">1</bibl> <bibtext> Gnanadoss, AA. 1960. Contracting Bernoullis iteration and recurrence relations. Mathematical Gazette, 44: 221–223.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref1" type="bt">2</bibl> <bibtext> Hardy, GH and Wright, EM. 1979. An Introduction to the Theory of Numbers, , 5th edn, Oxford: Oxford Science Publications.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref4" type="bt">3</bibl> <bibtext> Herstein, IN. 1975. Topics in Algebra, , 2nd edn, New Delhi: Wiley Eastern Limited.</bibtext> </blist> <blist> <bibl id="bib4" idref="ref7" type="bt">4</bibl> <bibtext> Householder, AS. 1970. The Numerical Treatment of a Single Non-linear Equations, New York: McGraw-Hill.</bibtext> </blist> <blist> <bibl id="bib5" idref="ref2" type="bt">5</bibl> <bibtext> Landau, E. 1927. Vorlesungen über Zahlentheorie, 1947New York: Verlag. von S. Hirzel, Leipzig, 1927; reprinted by Chelsea Publishing Co..</bibtext> </blist> <blist> <bibl id="bib6" idref="ref3" type="bt">6</bibl> <bibtext> Landau, E. 1999. Elementary Number Theory, Amer. Math. Soc., , 2nd edn, Providence, RI: AMS Chelsea Publishing.</bibtext> </blist> </ref> <aug> <p>By H. K. Pathak</p> <p>Reported by Author</p> </aug>
Header DbId: eric
DbLabel: ERIC
An: EJ781065
AccessLevel: 3
PubType: Academic Journal
PubTypeId: academicJournal
PreciseRelevancyScore: 0
IllustrationInfo
Items – Name: Title
  Label: Title
  Group: Ti
  Data: Some Problems and Conjectures in Number Theory
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Pathak%2C+H%2E+K%2E%22">Pathak, H. K.</searchLink>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22International+Journal+of+Mathematical+Education+in+Science+and+Technology%22"><i>International Journal of Mathematical Education in Science and Technology</i></searchLink>. Jan 2008 39(1):77-82.
– Name: Avail
  Label: Availability
  Group: Avail
  Data: Taylor & Francis, Ltd. 325 Chestnut Street Suite 800, Philadelphia, PA 19106. Tel: 800-354-1420; Fax: 215-625-2940; Web site: http://www.tandf.co.uk/journals/default.html
– Name: PeerReviewed
  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: PhysDesc
  Label: Physical Description
  Group: PhysDesc
  Data: PDF
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 6
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2008
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Descriptive
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Number+Concepts%22">Number Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Theories%22">Theories</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+Logic%22">Mathematical Logic</searchLink><br /><searchLink fieldCode="DE" term="%22Validity%22">Validity</searchLink><br /><searchLink fieldCode="DE" term="%22Equations+%28Mathematics%29%22">Equations (Mathematics)</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1080/00207390701607240
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0020-739X
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: In this article, we shall discuss some interesting, viable, meaningful, applicable and productive conjectures and methods to deal with some fundamental results in the theory of numbers.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: Author
– Name: Ref
  Label: Number of References
  Group: RefInfo
  Data: 6
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2007
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ781065
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ781065
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1080/00207390701607240
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 6
        StartPage: 77
    Subjects:
      – SubjectFull: Number Concepts
        Type: general
      – SubjectFull: Theories
        Type: general
      – SubjectFull: Mathematical Logic
        Type: general
      – SubjectFull: Validity
        Type: general
      – SubjectFull: Equations (Mathematics)
        Type: general
    Titles:
      – TitleFull: Some Problems and Conjectures in Number Theory
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Pathak, H. K.
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 01
              Type: published
              Y: 2008
          Identifiers:
            – Type: issn-print
              Value: 0020-739X
          Numbering:
            – Type: volume
              Value: 39
            – Type: issue
              Value: 1
          Titles:
            – TitleFull: International Journal of Mathematical Education in Science and Technology
              Type: main
ResultId 1