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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><ul class="breadcrumb"> <li> <a href="/topics/RecreationalMathematics.html">Recreational Mathematics</a> </li> <li> <a href="/topics/MathematicalRecords.html">Mathematical Records</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Terr.html">Terr</a> </li> </ul></nav>   <h1>Computational Number Theory</h1>  <hr class="margin-t-1-8 margin-b-3-4">   <div class="entry-content"> <p> Computational number theory is the branch of <a href="/NumberTheory.html">number theory</a> concerned with finding and implementing efficient computer algorithms for solving various problems in number theory. Much progress has been made in this field in recent years, both in terms of improved computer speed and in terms of finding more efficient algorithms. Two important applications of computational number theory are <a href="/PrimalityTest.html">primality testing</a> and <a href="/PrimeFactorization.html">prime factorization</a> of large integers. </p> <p> <a href="/PrimalityTest.html">Primality testing</a> is considered easy in the sense that very large general numbers (currently up to 4000 digits or so) can be tested reliably for primality. In fact, on August 6, 2002, Agrawal, Saxena, and Kayal found a <a href="/PolynomialTime.html">polynomial time</a> algorithm for testing and proving the primality of general numbers. Although this algorithm is still impractical, it was a landmark discovery, since <a href="/PolynomialTime.html">polynomial time</a> algorithms are considered easy. On the other hand, factoring is considered hard in the sense that no <a href="/PolynomialTime.html">polynomial time</a> algorithm is currently known for factoring integers. The largest general integer to be factored was RSA-576, a 174-digit number that is the product of two 87-digit primes. The fact that primality testing is easy but factoring is hard allows for secure encryption, such as <a href="/RSAEncryption.html">RSA encryption</a>. </p> <p> Other problems in computational number theory include computing the <a href="/GreatestCommonDivisor.html">greatest common divisor</a> of large numbers and computing various quantities associated with number fields, i.e., <a href="/ClassNumber.html">class numbers</a> and <a href="/ClassGroup.html">class groups</a>. </p> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">  <h2>See also</h2><a href="/AlgebraicNumberTheory.html">Algebraic Number Theory</a>, <a href="/ClassGroup.html">Class Group</a>, <a href="/ClassNumber.html">Class Number</a>, <a href="/NumberTheory.html">Number Theory</a>, <a href="/PrimalityTest.html">Primality Test</a>, <a href="/PrimeFactorization.html">Prime Factorization</a>, <a href="/RSAEncryption.html">RSA Encryption</a>, <a href="/RSANumber.html">RSA Number</a>     <p class="contributor"> <i>This entry contributed by <a target="_blank" href="/topics/Terr.html">David Terr</a></i> </p>   <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="https://www.wolframalpha.com/input/?i=aleph0">aleph0</a></li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=detect+regions+of+Saturn+image">detect regions of Saturn image</a></li> <li><a target="_blank" href="http://www.wolframalpha.com/input/?i=integrate+sin%5E2%28x%29+%2B+2+sin%5E4%282x%29+from+0+to+pi">integrate sin^2(x) + 2 sin^4(2x) from 0 to pi</a></li> </ul> </div> </div>   <h2>References</h2><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>Cohen, H. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387556400/ref=nosim/ericstreasuretro">A Course in Computational Algebraic Number Theory.</a></i> New York: Springer-Verlag, 1993.</cite><h2>Referenced on Wolfram|Alpha</h2><a href="http://www.wolframalpha.com/entities/mathworld/computational_number_theory/6n/zt/es/" title="Computational Number Theory" target="_blank">Computational Number Theory</a>   <h2>Cite this as:</h2> <p> <a href="/topics/Terr.html">Terr, David</a>. "Computational Number Theory." From <a href="/"><i>MathWorld</i></a>--A Wolfram Web Resource, created by <a href="/about/author.html">Eric W. Weisstein</a>. <a href="https://mathworld.wolfram.com/ComputationalNumberTheory.html">https://mathworld.wolfram.com/ComputationalNumberTheory.html</a> </p>  <h2>Subject classifications</h2><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><ul class="breadcrumb"> <li> <a href="/topics/RecreationalMathematics.html">Recreational Mathematics</a> </li> <li> <a href="/topics/MathematicalRecords.html">Mathematical Records</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Terr.html">Terr</a> </li> </ul></nav>  </div> </section> </section>  </div> </main> <aside id="bottom"> <style> #bottom { padding-bottom: 65px; } #acknowledgment { display:none; } .attribution { font-size: .75rem; font-style: italic; } footer ul li:not(:last-of-type)::after { background: #a3a3a3; margin-left: .3rem; margin-right: .1rem; } @media all and (max-width: 900px) { .attribution { font-size: 12px; } } @media (max-width: 600px) { footer { max-width: 360px; } footer ul { max-width: 360px; } footer ul:nth-child(1) li:nth-child(2):after { content: ""; height: 11px; } footer ul:nth-child(1) li:nth-child(3):after { content: ""; height: 0px; } } </style> <footer> <ul> <li><a href="/about/">About MathWorld</a></li> <li><a href="/classroom/">MathWorld Classroom</a></li> <li><a href="/contact/">Contribute</a></li> <li><a href="https://www.amazon.com/exec/obidos/ASIN/1420072218/ref=nosim/weisstein-20" target="_blank">MathWorld Book</a></li> <li class="display-n display-ib__600"><a href="https://www.wolfram.com" target="_blank">wolfram.com</a></li> </ul> <ul> <li class="display-n__600"><a href="/whatsnew/">13,208 Entries</a></li> <li class="display-n__600"><a href="/whatsnew/">Last Updated: Thu Nov 21 2024</a></li>  <li><a href="https://www.wolfram.com" target="_blank">©1999–2024 Wolfram Research, Inc.</a></li> <li><a href="https://www.wolfram.com/legal/terms/mathworld.html" target="_blank">Terms of Use</a></li> </ul> <ul class="wolfram"> <li class="display-n__600 display-n__900"><a href="https://www.wolfram.com" target="_blank" aria-label="Wolfram"><img src="/images/footer/wolfram-logo.png" alt="Wolfram" title="Wolfram" width="121" height="28"></a></li> <li class="display-n__600"><a href="https://www.wolfram.com" target="_blank">wolfram.com</a></li> <li class="display-n__600"><a href="https://www.wolfram.com/education/" target="_blank">Wolfram for Education</a></li> <li class="attribution">Created, developed and nurtured by Eric Weisstein at Wolfram Research</li> </ul> </footer> <section id="acknowledgment"> <i>Created, developed and nurtured by Eric Weisstein at Wolfram Research</i> </section> </aside> <script type="text/javascript" src="/scripts/scripts.js"></script> <script src="/common/js/c2c/1.0/WolframC2C.js"></script> <script src="/common/js/c2c/1.0/WolframC2CGui.js"></script> <script src="/common/js/c2c/1.0/WolframC2CDefault.js"></script> <link rel="stylesheet" href="/common/js/c2c/1.0/WolframC2CGui.css.en"> <style> .wolfram-c2c-wrapper { padding: 0px !important; border: 0px; } .wolfram-c2c-wrapper:active { border: 0px; } .wolfram-c2c-wrapper:hover { border: 0px; } </style> <script> let c2cWrittings = new WolframC2CDefault({'triggerClass':'mathworld-c2c_above', 'uniqueIdPrefix': 'mathworld-c2c_above-'}); </script> <style> #IPstripe-outer { background: #47a2af; } #IPstripe-outer:hover { background: #0095aa; } </style> <div id="IPstripe-wrap"></div> <script src="/common/stripe/stripe.en.js"></script> </body> </html>