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<!doctype html> <html lang="en" class="numbertheory"> <head> <title>Number Theory -- from Wolfram MathWorld</title> <meta name="DC.Title" content="Number Theory" /> <meta name="DC.Creator" content="Weisstein, Eric W." /> <meta name="DC.Description" content="Number theory is a vast and fascinating field of mathematics, sometimes called "higher arithmetic," consisting of the study of the properties of whole numbers. Primes and prime factorization are especially important in number theory, as are a number of functions such as the divisor function, Riemann zeta function, and totient function. Excellent introductions to number theory may be found in Ore (1988) and Beiler (1966). The classic history on the subject (now slightly dated) is..." /> <meta name="description" content="Number theory is a vast and fascinating field of mathematics, sometimes called "higher arithmetic," consisting of the study of the properties of whole numbers. 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method="get" action="/search/" name="search" id="search-mobile" accept-charset="UTF-8"> <input type="text" name="query" placeholder="Search" id="searchFieldMobile"> <img src="/images/header/search-icon.png" width="18" height="18" id="mobile-search" class="search-btn" title="Search" alt="Search" onclick="submitForm()"> </form> <nav class="topics-nav"> <a href="/topics/Algebra.html" id="sidebar-algebra"> Algebra </a> <a href="/topics/AppliedMathematics.html" id="sidebar-appliedmathematics"> Applied Mathematics </a> <a href="/topics/CalculusandAnalysis.html" id="sidebar-calculusandanalysis"> Calculus and Analysis </a> <a href="/topics/DiscreteMathematics.html" id="sidebar-discretemathematics"> Discrete Mathematics </a> <a href="/topics/FoundationsofMathematics.html" id="sidebar-foundationsofmathematics"> Foundations of Mathematics </a> <a href="/topics/Geometry.html" id="sidebar-geometry"> Geometry </a> <a href="/topics/HistoryandTerminology.html" id="sidebar-historyandterminology"> History and Terminology </a> <a href="/topics/NumberTheory.html" id="sidebar-numbertheory"> Number Theory </a> <a href="/topics/ProbabilityandStatistics.html" id="sidebar-probabilityandstatistics"> Probability and Statistics </a> <a href="/topics/RecreationalMathematics.html" id="sidebar-recreationalmathematics"> Recreational Mathematics </a> <a href="/topics/Topology.html" id="sidebar-topology"> Topology </a> </nav> <nav class="secondary-nav"> <a href="/letters/"> Alphabetical Index </a> <a href="/whatsnew/"> New in MathWorld </a> </nav> </section> <section id="content">  <nav class="breadcrumbs"><ul class="breadcrumb"> <li> <a href="/topics/NumberTheory.html">Number Theory</a> </li> <li> <a href="/topics/GeneralNumberTheory.html">General Number Theory</a> </li> </ul></nav>   <h1>Number Theory</h1>  <hr class="margin-t-1-8 margin-b-3-4">   <div class="entry-content"> <p> Number theory is a vast and fascinating field of mathematics, sometimes called "higher arithmetic," consisting of the study of the properties of whole numbers. <a href="/PrimeNumber.html">Primes</a> and <a href="/PrimeFactorization.html">prime factorization</a> are especially important in number theory, as are a number of functions such as the <a href="/DivisorFunction.html">divisor function</a>, <a href="/RiemannZetaFunction.html">Riemann zeta function</a>, and <a href="/TotientFunction.html">totient function</a>. Excellent introductions to number theory may be found in Ore (1988) and Beiler (1966). The classic history on the subject (now slightly dated) is that of Dickson (2005abc). </p> <p> The great difficulty in proving relatively simple results in number theory prompted no less an authority than Gauss to remark that "it is just this which gives the higher arithmetic that magical charm which has made it the favorite science of the greatest mathematicians, not to mention its inexhaustible wealth, wherein it so greatly surpasses other parts of mathematics." Gauss, often known as the "prince of mathematics," called mathematics the "queen of the sciences" and considered number theory the "queen of mathematics" (Beiler 1966, Goldman 1997). </p> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">  <h2>See also</h2><a href="/AbstractAlgebra.html">Abstract Algebra</a>, <a href="/AdditiveNumberTheory.html">Additive Number Theory</a>, <a href="/AlgebraicNumberTheory.html">Algebraic Number Theory</a>, <a href="/AnalyticNumberTheory.html">Analytic Number Theory</a>, <a href="/Arithmetic.html">Arithmetic</a>, <a href="/ComputationalNumberTheory.html">Computational Number Theory</a>, <a href="/Congruence.html">Congruence</a>, <a href="/DiophantineEquation.html">Diophantine Equation</a>, <a href="/DivisorFunction.html">Divisor Function</a>, <a href="/ElementaryNumberTheory.html">Elementary Number Theory</a>, <a href="/GoedelsFirstIncompletenessTheorem.html">Gödel's First Incompleteness Theorem</a>, <a href="/GoedelsSecondIncompletenessTheorem.html">Gödel's Second Incompleteness Theorem</a>, <a href="/MultiplicativeNumberTheory.html">Multiplicative Number Theory</a>, <a href="/NumberTheoreticFunction.html">Number Theoretic Function</a>, <a href="/PeanosAxioms.html">Peano's Axioms</a>, <a href="/PrimeCountingFunction.html">Prime Counting Function</a>, <a href="/PrimeFactorization.html">Prime Factorization</a>, <a href="/PrimeNumber.html">Prime Number</a>, <a href="/QuadraticReciprocityTheorem.html">Quadratic Reciprocity Theorem</a>, <a href="/RiemannZetaFunction.html">Riemann Zeta Function</a>, <a href="/TotientFunction.html">Totient Function</a> <a href="/classroom/NumberTheory.html" class="explore-classroom">Explore this topic in the MathWorld classroom</a>       <h2>Explore with Wolfram|Alpha</h2> <div id="WAwidget"> <div class="WAwidget-wrapper"> <img alt="WolframAlpha" title="WolframAlpha" src="/images/wolframalpha/WA-logo.png" width="136" height="20"> <form name="wolframalpha" action="https://www.wolframalpha.com/input/" target="_blank"> <input type="text" name="i" class="search" placeholder="Solve your math problems and get step-by-step solutions" value=""> <button type="submit" title="Evaluate on WolframAlpha"></button> </form> </div> <div class="WAwidget-wrapper try"> <p class="text-align-r"> More things to try: </p> <ul> <li> <a target="_blank" href="http://www.wolframalpha.com/input/?i=number+theory"> number theory </a> </li> <li> <a target="_blank" href="http://www.wolframalpha.com/input/?i=largest+prime+number"> largest prime number </a> </li> <li> <a target="_blank" href="http://www.wolframalpha.com/input/?i=Moebius+function%28n%29"> Moebius function(n) </a> </li> </ul> </div> </div>   <h2>References</h2><cite>Andrews, G. E. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486682528/ref=nosim/ericstreasuretro">Number Theory.</a></i> New York: Dover, 1994.</cite><cite>Andrews, G. E.; Berndt, B. C.; and Rankin, R. A. (Ed.). <i><a href="http://www.amazon.com/exec/obidos/ASIN/012058560X/ref=nosim/ericstreasuretro">Ramanujan Revisited: Proceedings of the Centenary Conference, University of Illinois at Urbana-Champaign, June 1-5, 1987.</a></i> Boston, MA: Academic Press, 1988.</cite><cite>Anglin, W. S. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0792332873/ref=nosim/ericstreasuretro">The Queen of Mathematics: An Introduction to Number Theory.</a></i> Dordrecht, Netherlands: Kluwer, 1995.</cite><cite>Apostol, T. M. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387901639/ref=nosim/ericstreasuretro">Introduction to Analytic Number Theory.</a></i> New York: Springer-Verlag, 1976.</cite><cite>Ayoub, R. G. <i><a href="http://www.amazon.com/exec/obidos/ASIN/B0006AYL8Q/ref=nosim/ericstreasuretro">An Introduction to the Analytic Theory of Numbers.</a></i> Providence, RI: Amer. Math. Soc., 1963.</cite><cite>Beiler, A. H. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486210960/ref=nosim/ericstreasuretro">Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, 2nd ed.</a></i> New York: Dover, 1966.</cite><cite>Bellman, R. E. <i><a href="http://www.amazon.com/exec/obidos/ASIN/080530360X/ref=nosim/ericstreasuretro">Analytic Number Theory: An Introduction.</a></i> Reading, MA: Benjamin/Cummings, 1980.</cite><cite>Berndt, B. C. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387961100/ref=nosim/ericstreasuretro">Ramanujan's Notebooks, Part I.</a></i> New York: Springer-Verlag, 1985.</cite><cite>Berndt, B. C. <i><a href="http://www.amazon.com/exec/obidos/ASIN/038796794X/ref=nosim/ericstreasuretro">Ramanujan's Notebooks, Part II.</a></i> New York: Springer-Verlag, 1988.</cite><cite>Berndt, B. C. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387975039/ref=nosim/ericstreasuretro">Ramanujan's Notebooks, Part III.</a></i> New York: Springer-Verlag, 1997a.</cite><cite>Berndt, B. C. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387941096/ref=nosim/ericstreasuretro">Ramanujan's Notebooks, Part IV.</a></i> New York: Springer-Verlag, 1993.</cite><cite>Berndt, B. C. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387949410/ref=nosim/ericstreasuretro">Ramanujan's Notebooks, Part V.</a></i> New York: Springer-Verlag, 1997b.</cite><cite>Berndt, B. C. and Rankin, R. A. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0821802879/ref=nosim/ericstreasuretro">Ramanujan: Letters and Commentary.</a></i> Providence, RI: Amer. Math. Soc, 1995.</cite><cite>Borwein, J. M. and Borwein, P. B. <i><a href="http://www.amazon.com/exec/obidos/ASIN/047131515X/ref=nosim/ericstreasuretro">Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.</a></i> New York: Wiley, 1987.</cite><cite>Bressoud, D. M. and Wagon, S. <i><a href="http://www.amazon.com/exec/obidos/ASIN/1930190107/ref=nosim/ericstreasuretro">A Course in Computational Number Theory.</a></i> London: Springer-Verlag, 2000.</cite><cite>Burr, S. A. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0821855018/ref=nosim/ericstreasuretro">The Unreasonable Effectiveness of Number Theory.</a></i> Providence, RI: Amer. Math. Soc., 1992.</cite><cite>Burton, D. M. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0070094667/ref=nosim/ericstreasuretro">Elementary Number Theory, 4th ed.</a></i> Boston, MA: Allyn and Bacon, 1989.</cite><cite>Carmichael, R. D. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486438031/ref=nosim/ericstreasuretro">The Theory of Numbers, and Diophantine Analysis.</a></i> New York: Dover, 1959.</cite><cite>Cohen, H. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387987274/ref=nosim/ericstreasuretro">Advanced Topics in Computational Number Theory.</a></i> New York: Springer-Verlag, 2000.</cite><cite>Cohn, H. <i><a href="http://www.amazon.com/exec/obidos/ASIN/048664023X/ref=nosim/ericstreasuretro">Advanced Number Theory.</a></i> New York: Dover, 1980.</cite><cite>Courant, R. and Robbins, H. "The Theory of Numbers." Supplement to Ch. 1 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0195105192/ref=nosim/ericstreasuretro">What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.</a></i> Oxford, England: Oxford University Press, pp. 21-51, 1996.</cite><cite>Davenport, H. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521422272/ref=nosim/ericstreasuretro">The Higher Arithmetic: An Introduction to the Theory of Numbers, 6th ed.</a></i> Cambridge, England: Cambridge University Press, 1992.</cite><cite>Davenport, H. and Montgomery, H. L. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387905332/ref=nosim/ericstreasuretro">Multiplicative Number Theory, 2nd ed.</a></i> New York: Springer-Verlag, 1980.</cite><cite>Dickson, L. E. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486442322/ref=nosim/ericstreasuretro">History of the Theory of Numbers, Vol. 1: Divisibility and Primality.</a></i> New York: Dover, 2005a.</cite><cite>Dickson, L. E. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486442330/ref=nosim/ericstreasuretro">History of the Theory of Numbers, Vol. 2: Diophantine Analysis.</a></i> New York: Dover, 2005b.</cite><cite>Dickson, L. E. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486442349/ref=nosim/ericstreasuretro">History of the Theory of Numbers, Vol. 3: Quadratic and Higher Forms.</a></i> New York: Dover, 2005c.</cite><cite>Dudley, U. <i><a href="http://www.amazon.com/exec/obidos/ASIN/071670076X/ref=nosim/ericstreasuretro">Elementary Number Theory.</a></i> San Francisco, CA: W. H. Freeman, 1978.</cite><cite>Friedberg, R. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486281337/ref=nosim/ericstreasuretro">An Adventurer's Guide to Number Theory.</a></i> New York: Dover, 1994.</cite><cite>Gauss, C. F. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387962549/ref=nosim/ericstreasuretro">Disquisitiones Arithmeticae.</a></i> New Haven, CT: Yale University Press, 1966.</cite><cite>Goldman, J. R. <i><a href="http://www.amazon.com/exec/obidos/ASIN/1568810067/ref=nosim/ericstreasuretro">The Queen of Mathematics: An Historically Motivated Guide to Number Theory.</a></i> Wellesley, MA: A K Peters, 1997.</cite><cite>Guy, R. K. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387208607/ref=nosim/ericstreasuretro">Unsolved Problems in Number Theory, 3rd ed.</a></i> New York: Springer-Verlag, 2004.</cite><cite>Hardy, G. H. and Wright, E. M. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0198531710/ref=nosim/ericstreasuretro">An Introduction to the Theory of Numbers, 5th ed.</a></i> Oxford, England: Clarendon Press, 1979.</cite><cite>Hardy, G. H. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0821820230/ref=nosim/ericstreasuretro">Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed.</a></i> New York: Chelsea, 1959.</cite><cite>Hasse, H. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387082751/ref=nosim/ericstreasuretro">Number Theory.</a></i> Berlin: Springer-Verlag, 1980.</cite><cite>Herkommer, M. A. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0079130747/ref=nosim/ericstreasuretro">Number Theory: A Programmer's Guide.</a></i> New York: McGraw-Hill, 1999.</cite><cite>Ireland, K. F. and Rosen, M. I. <i><a href="http://www.amazon.com/exec/obidos/ASIN/038797329X/ref=nosim/ericstreasuretro">A Classical Introduction to Modern Number Theory, 2nd ed.</a></i> New York: Springer-Verlag, 1995.</cite><cite>Kato, K.; Kurokawa, N.; and Saito, T. <i><a href="http://www.amazon.com/exec/obidos/ASIN/082180863X/ref=nosim/ericstreasuretro">Number Theory 1: Fermat's Dream.</a></i> Providence, RI: Amer. Math. Soc., 2000.</cite><cite>Klee, V. and Wagon, S. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0883853159/ref=nosim/ericstreasuretro">Old and New Unsolved Problems in Plane Geometry and Number Theory.</a></i> Washington, DC: Math. Assoc. Amer., 1991.</cite><cite>Koblitz, N. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387942939/ref=nosim/ericstreasuretro">A Course in Number Theory and Cryptography.</a></i> New York: Springer-Verlag, 1987.</cite><cite>Landau, E. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0821820044/ref=nosim/ericstreasuretro">Elementary Number Theory, 2nd ed.</a></i> New York: Chelsea, 1999.</cite><cite>Lang, S. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387942254/ref=nosim/ericstreasuretro">Algebraic Number Theory, 2nd ed.</a></i> New York: Springer-Verlag, 1994.</cite><cite>Lenstra, H. W. and Tijdeman, R. (Eds.). <i>Computational Methods in Number Theory, 2 vols.</i> Amsterdam: Mathematisch Centrum, 1982.</cite><cite>LeVeque, W. J. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486689069/ref=nosim/ericstreasuretro">Fundamentals of Number Theory.</a></i> New York: Dover, 1996.</cite><cite>MathPages. "Number Theory." <a href="http://www.mathpages.com/home/inumber.htm">http://www.mathpages.com/home/inumber.htm</a>.</cite><cite>Mitrinović, D. S. and Sándor, J. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0792338235/ref=nosim/ericstreasuretro">Handbook of Number Theory.</a></i> Dordrecht, Netherlands: Kluwer, 1995.</cite><cite>Mollin, R. A. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0849339898/ref=nosim/ericstreasuretro">Algebraic Number Theory.</a></i> Boca Raton, FL: CRC Press, 1999.</cite><cite>Mollin, R. A. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0849339871/ref=nosim/ericstreasuretro">Fundamental Number Theory with Applications.</a></i> Boca Raton, FL: CRC Press, 1998.</cite><cite>Niven, I. M.; Zuckerman, H. S.; and Montgomery, H. L. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0471625469/ref=nosim/ericstreasuretro">An Introduction to the Theory of Numbers, 5th ed.</a></i> New York: Wiley, 1991.</cite><cite>Ogilvy, C. S. and Anderson, J. T. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486257789/ref=nosim/ericstreasuretro">Excursions in Number Theory.</a></i> New York: Dover, 1988.</cite><cite>Ore, Ø. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0883856204/ref=nosim/ericstreasuretro">Invitation to Number Theory.</a></i> Washington, DC: Math. Assoc. Amer., 1967.</cite><cite>Ore, Ø. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486656209/ref=nosim/ericstreasuretro">Number Theory and Its History.</a></i> New York: Dover, 1988.</cite><cite>Rose, H. E. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0198523769/ref=nosim/ericstreasuretro">A Course in Number Theory, 2nd ed.</a></i> Oxford, England: Clarendon Press, 1995.</cite><cite>Rosen, K. H. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0201578891/ref=nosim/ericstreasuretro">Elementary Number Theory and Its Applications, 3rd ed.</a></i> Reading, MA: Addison-Wesley, 1993.</cite><cite>Schroeder, M. R. <i><a href="http://www.amazon.com/exec/obidos/ASIN/3540620060/ref=nosim/ericstreasuretro">Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, 3rd ed.</a></i> New York: Springer-Verlag, 1997.</cite><cite>Shanks, D. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0828412979/ref=nosim/ericstreasuretro">Solved and Unsolved Problems in Number Theory, 4th ed.</a></i> New York: Chelsea, 1993.</cite><cite>Sierpinski, W. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0444000712/ref=nosim/ericstreasuretro">250 Problems in Elementary Number Theory.</a></i> New York: American Elsevier, 1970.</cite><cite>Uspensky, J. V. and Heaslet, M. A. <i><a href="http://www.amazon.com/exec/obidos/ASIN/B000AQI3BM/ref=nosim/ericstreasuretro">Elementary Number Theory.</a></i> New York: McGraw-Hill, 1939.</cite><cite>Vinogradov, I. M. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486495302/ref=nosim/ericstreasuretro">Elements of Number Theory, 5th rev. ed.</a></i> New York: Dover, 1954.</cite><cite>Weil, A. <i><a href="http://www.amazon.com/exec/obidos/ASIN/3540586555/ref=nosim/ericstreasuretro">Basic Number Theory, 3rd ed.</a></i> Berlin: Springer-Verlag, 1995.</cite><cite>Weil, A. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0817631410/ref=nosim/ericstreasuretro">Number Theory: An Approach Through History From Hammurapi to Legendre.</a></i> Boston, MA: Birkhäuser, 1984.</cite><cite>Weisstein, E. W. "Books about Number Theory." <a href="http://www.ericweisstein.com/encyclopedias/books/NumberTheory.html">http://www.ericweisstein.com/encyclopedias/books/NumberTheory.html</a>.</cite><cite>Weyl, H. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0691059179/ref=nosim/ericstreasuretro">Algebraic Theory of Numbers.</a></i> Princeton, NJ: Princeton University Press, 1998.</cite><cite>Yildirim, C. Y. and Stepanov, S. A. (Eds.). <i><a href="http://www.amazon.com/exec/obidos/ASIN/0824719697/ref=nosim/ericstreasuretro">Number Theory and Its Applications.</a></i> New York: Dekker, 1998.</cite><cite>Young, J. W. A. "The Theory of Numbers." Ch. 7 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/B0006AG87I/ref=nosim/ericstreasuretro">Monographs on Topics of Modern Mathematics Relevant to the Elementary Field</a></i> (Ed. J. W. A. Young). New York: Dover, pp. 306-349, 1955.</cite><h2>Referenced on Wolfram|Alpha</h2><a href="http://www.wolframalpha.com/entities/mathworld/number_theory/vu/5q/76/" title="Number Theory" target="_blank">Number Theory</a>   <h2>Cite this as:</h2> <p> <a href="/about/author.html">Weisstein, Eric W.</a> "Number Theory." 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