ࡱ> q ubjbjt+t+ AAq]$bV V V V V "x GIIIIII$8,dm V V mV V  V V G G'GV 4`@}[Kx~N?Number Theory Links for Rosens Elem. Number Theory and Applications Numbers, Sequences, and Sums Page 8 To learn more about figurate number consult  HYPERLINK http://mathworld.wolfram.com/FigurateNumber.html http://mathworld.wolfram.com/FigurateNumber.html Page 9 You can use Neil Sloane's On-Line Encyclopedia of Integer Sequences Web site to determine the possible identity of an integer sequence from its first few terms. You can also check out some "hot sequences" and puzzle sequences that are quite challenging.  HYPERLINK http://www.research.att.com/~njas/sequences/index.html http://www.research.att.com/~njas/sequences/index.html 1.2 Numbers, Sequences, and Sums Page 23 A picture of the 19th century original box cover for the Tower of Hanoi puzzle and the text of the original instructions in French, and translated into English, can be seen at the site of Paul K. Stockmeyer, a computer science professor at William and Mary College at  HYPERLINK http://www.cs.wm.edu/~pkstoc/toh.html http://www.cs.wm.edu/~pkstoc/toh.html Several interesting papers about the Tower of Hanoi problem and its generalizations written by Paul K. Stockmeyer can be downloaded from  HYPERLINK http://www.cs.wm.edu/~pkstoc/h_papers.html http://www.cs.wm.edu/~pkstoc/h_papers.html 1.3 The Fibonacci Numbers Page 25 A variety of ways that Fibonacci number arise in nature, including counting rabbits, can be found on Ron Knott's page at the Department of Computing, University of Surrey site:  HYPERLINK http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html 1.4 Divisibility Page 37 You can find the excellent expository article by Jeff Lagarias on the Collatz conjecture, which goes by many different names, including the "3x+1 problem" at  HYPERLINK http://www.cecm.sfu.ca/organics/papers/lagarias/index.html http://www.cecm.sfu.ca/organics/papers/lagarias/index.html 2.1 Representations of Integers Page 47 Interesting information concerning Kaprekar's constant and related topics can be found at  HYPERLINK http://mathworld.wolfram.com/KaprekarRoutine.html http://mathworld.wolfram.com/KaprekarRoutine.html 3.1 Prime Numbers Page 66 A wealth of information about primes can be found at the Prime Page  HYPERLINK http://www.utm.edu/research/primes/ http://www.utm.edu/research/primes/ An excellent survey about prime numbers can be found at the St Andrews History of Mathematics Archive at  HYPERLINK http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Prime_numbers.html http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Prime_numbers.html Page 67 Biographical information about Eratosthenes can be found at the MacTutor History of Mathematics Archive at  HYPERLINK http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Eratosthenes.html http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Eratosthenes.html You can find an applets for running the sieve of Eratosthenes at  HYPERLINK http://www.math.utah.edu/~alfeld/math/prime.html http://www.math.utah.edu/~alfeld/math/prime.html Information about the sieve of Eratosthenes and links for implementations in C, Java, and Perl can be found at  HYPERLINK http://www.utm.edu/research/primes/programs/Eratosthenes http://www.utm.edu/research/primes/programs/Eratosthenes Page 75 A description of some of the open questions about primes can be found on the Prime Pages at  HYPERLINK http://www.utm.edu/research/primes/notes/conjectures http://www.utm.edu/research/primes/notes/conjectures You can learn more about the twin prime conjecture and related conjectures at  HYPERLINK http://www.ltkz.demon.co.uk/ktuplets.htm http://www.ltkz.demon.co.uk/ktuplets.htm Information about the current status of numerical evidence supporting Golbach's conjecture can be found at  HYPERLINK http://www.informatik.uni-giessen.de/staff/richstein/ca/Goldbach.html http://www.informatik.uni-giessen.de/staff/richstein/ca/Goldbach.html 3.2 Greatest Common Divisors Page 80 An excellent starting place for learning about greatest common divisors is  HYPERLINK http://www.utm.edu/research/primes/glossary/GCD.html http://www.utm.edu/research/primes/glossary/GCD.html Page 85 Information about Farey series and about Farey himself can be found at  HYPERLINK http://www.cut-the-knot.com/blue/Farey.html http://www.cut-the-knot.com/blue/Farey.html Programs for computing Farey series can be found at  HYPERLINK http://www.maths.usyd.edu.au:8000/u/magma/Examples/node6.html http://www.maths.usyd.edu.au:8000/u/magma/Examples/node6.html 3.3 The Euclidean Algorithm Page 92 You can find source code for a C program implementing the extended Euclidean algorithm at  HYPERLINK http://www.mindspring.com/~pate/crypto/chap02.html http://www.mindspring.com/~pate/crypto/chap02.html 3.5 Factorization Methods and the Fermat Numbers A succinct report on modern factorization methods can be found as part of the RSA Labs FAQ at  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/2-3-4.html http://www.rsasecurity.com/rsalabs/faq/2-3-4.html Another good place to learn about different factorization methods is in Eric Weisstein's World of Mathematics at  HYPERLINK http://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html http://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html Page 113 You can learn about the RSA Factoring Challenge at the following Web page at the RSA Data Security site:  HYPERLINK http://www.rsasecurity.com/rsalabs/challenges/factoring/ http://www.rsasecurity.com/rsalabs/challenges/factoring/ Page 114 The currently known data about the factorization of Fermat numbers can be found on the following page created and maintained by Wilfrid Keller  HYPERLINK http://vamri.xray.ufl.edu/proths/fermat.html http://vamri.xray.ufl.edu/proths/fermat.html Cash prizes are offered for finding prime factors of certain Fermat numbers. See  HYPERLINK http://www.perfsci.com/prizes.html http://www.perfsci.com/prizes.html Page 115 Information about the Cunningham project, including the "most wanted" numbers awaiting factorization can be found at  HYPERLINK http://www.cerias.purdue.edu/homes/ssw/cun/ http://www.cerias.purdue.edu/homes/ssw/cun/ 3.6 Linear Diophantine Equations Page 120 An interactive applet for solving linear diophantine equations can be found at  HYPERLINK http://www.thoralf2.uwaterloo.ca/htdocs/linear.html http://www.thoralf2.uwaterloo.ca/htdocs/linear.html 4.1 Introduction to Congruences Page 127 Modular arithmetic is discussed at the Cut-the-Knot site at  HYPERLINK http://www.cut-the-knot.com/blue/Modulo.html http://www.cut-the-knot.com/blue/Modulo.html You can find source code for a C program implementing the algorithm for modular exponentiation by repeated squaring and multiplying at  HYPERLINK http://www.mindspring.com/~pate/crypto/chap02.html http://www.mindspring.com/~pate/crypto/chap02.html 4.2 Linear Congruences Page 139 You can find an approach for solving linear congruences at  HYPERLINK http://www.unc.edu/~rowlett/Math81/notes/Oct06.html http://www.unc.edu/~rowlett/Math81/notes/Oct06.html Page 140 You can find source code for a C program that computes modular inverses at  HYPERLINK http://www.mindspring.com/~pate/crypto/chap02.html http://www.mindspring.com/~pate/crypto/chap02.html 4.3 The Chinese Remainder Theorem Page 143 You can learn more about the Chinese remainder theorem at  HYPERLINK http://www.cut-the-knot.com/blue/chinese.html http://www.cut-the-knot.com/blue/chinese.html You can calculate your biorhythms using the program at  HYPERLINK http://www.facade.com/biorhythm http://www.facade.com/biorhythm 4.5 Systems of Linear Congruences Page 169 An excellent starting point for exploring Magic squares on the Web is  HYPERLINK http://forum.swarthmore.edu/alejandre/magic.square.html http://forum.swarthmore.edu/alejandre/magic.square.html 4.6 Factoring Using the Pollard Rho Method Page 170 Source code for the Pollard rho factorization method can be found at  HYPERLINK http://www.mindspring.com/~pate/course/chap08.html http://www.mindspring.com/~pate/course/chap08.html 5.1 Divisibility Tests Page 173 You can find a clear explanation, written by Robert L. Ward, why different divisibility tests work for numbers in decimal notation at  HYPERLINK http://forum.swarthmore.edu/k12/mathtips/ward.html http://forum.swarthmore.edu/k12/mathtips/ward.html Page 177 Information on repunits, including the repunits known to be prime, can be found at  HYPERLINK http://www.utm.edu/research/primes/glossary/Repunit.html http://www.utm.edu/research/primes/glossary/Repunit.html 5.2 The Perpetual Calendar Page 179 A perpetual calendar implemented by Michael Bertrand in Java can be found at  HYPERLINK http://www.execpc.com/~mikeber/calendar.html http://www.execpc.com/~mikeber/calendar.html 5.3 Round-Robin Tournaments Page 184 An interesting algorithm for scheduling double round-robin tournaments where each team plays each other twice so that each time plays at home, and so that a variety of constraints can be met, can be found at  HYPERLINK http://www.comp.nus.edu.sg/~henz/projects/acc/index.html http://www.comp.nus.edu.sg/~henz/projects/acc/index.html 5.4 Hashing Functions Page 186 A definition of hashing functions can be found at  HYPERLINK http://www.whatis.com/WhatIs_Definition_Page/0,4152,212230,00.html http://www.whatis.com/WhatIs_Definition_Page/0,4152,212230,00.html A discussion of hashing functions from a cryptographic standpoint is available as part of the RSA Laboratories cryptography FAQ at  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/2-1-6.html http://www.rsasecurity.com/rsalabs/faq/2-1-6.html 5.5 Check Digits Page 191 A comprehensive discussion of various schemes used to construct check digits can be found at  HYPERLINK http://www.augustana.ab.ca/~mohrj/algorithms/checkdigit.html http://www.augustana.ab.ca/~mohrj/algorithms/checkdigit.html Page 192 You can find a Java applet for computing the check digits for ISBNs at  HYPERLINK http://www.albar.co.uk/tooltips.htm#isbncdc http://www.albar.co.uk/tooltips.htm#isbncdc Page 195 You can find a Java applet for computing check digits for UPCs at  HYPERLINK http://www.albar.co.uk/tooltips.htm#upcacdc http://www.albar.co.uk/tooltips.htm#upcacdc 7.3 Perfect Numbers and Mersenne Primes Page 239 A survey article about perfect numbers can be found at the St. Andrews History of Mathematics site at  HYPERLINK http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Perfect_numbers.html http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Perfect_numbers.html Page 241 A wealth of information about Mersenne primes can be found at  HYPERLINK http://www.utm.edu/research/primes/mersenne.shtml http://www.utm.edu/research/primes/mersenne.shtml Page 244 The history of the search for Mersenne primes is described in detail at  HYPERLINK http://www.utm.edu/research/primes/mersenne.shtml http://www.utm.edu/research/primes/mersenne.shtml Luke Welsh, one of the discoverers of the 29th Mersenne prime, has an excellent site containing a wealth of information about Marin Mersenne and the search for Mersenne primes at  HYPERLINK http://www.scruznet.com/~luke/mersenne.htm http://www.scruznet.com/~luke/mersenne.htm Page 245 You can learn about the Great Internet Mersenne Prime Search, download software to look for new Mersenne primes, and join the search itself at  HYPERLINK http://www.mersenne http://www.mersenne.org You can learn about progress with the search for Mersenne primes over PrimeNet and obtain software for joining PrimeNet which is associated with the Great Internet Prime Search at  HYPERLINK http://www.entropia.com/ips/ http://www.entropia.com/ips/ Page 248 To learn more about amicable numbers you can consult  HYPERLINK http://xraysgi.ims.uconn.edu/amicable.html http://xraysgi.ims.uconn.edu/amicable.html To access a list of all known amicable pairs go to  HYPERLINK http://www.vejlehs.dk/staff/jmp/aliquot/knwnap.htm http://www.vejlehs.dk/staff/jmp/aliquot/knwnap.htm Information about multiply perfect numbers can be found at  HYPERLINK http://www.uni-bielefeld.de/~achim/mpn.html http://www.uni-bielefeld.de/~achim/mpn.html 8.1 Character Ciphers Page 260 You can learn more about cryptography by checking out the Frequently Asked Questions (FAQs) on cryptography at  HYPERLINK http://www.faqs.org/faqs/cryptography-faq/ http://www.faqs.org/faqs/cryptography-faq/ RSA Laboratories has compiled their own highly informative set of Frequently Asked Questions (FAQs) in cryptography which are accessible at  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/ http://www.rsasecurity.com/rsalabs/faq/ The basic terminology and concepts of cryptography is described on the page  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/1-2.html http://www.rsasecurity.com/rsalabs/faq/1-2.html An excellent introduction to cryptographic terminology and concepts can be found at  HYPERLINK http://www.ssh.fi/tech/crypto/intro.html http://www.ssh.fi/tech/crypto/intro.html 8.2 Block and Stream Ciphers Page 268 The basic concept of a block cipher is described in detail at  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/2-1-4.html http://www.rsasecurity.com/rsalabs/faq/2-1-4.html Page 269 An excellent description of the Vigenere cipher can be found at  HYPERLINK http://www.trincoll.edu/depts/cpsc/cryptography/vigenere http://www.trincoll.edu/depts/cpsc/cryptography/vigenere An on-line program for cryptanalysis of ciphertext encrypted using the Vigenere cipher is available at:  HYPERLINK http://www.cs.arizona.edu/people/muth/Cipher/query_vcb http://www.cs.arizona.edu/people/muth/Cipher/query_vcb Page 274 The complete specification of the DES is available from the National Institute of Standards and Technology (NIST) at  HYPERLINK http://www.itl.nist.gov/fipspubs/fip46-2.htm http://www.itl.nist.gov/fipspubs/fip46-2.htm You can learn more about the DES by consulting  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/3-2-1 http://www.rsasecurity.com/rsalabs/faq/3-2-1 Page 275 You can learn more about the AES by consulting  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/3-3-1.html http://www.rsasecurity.com/rsalabs/faq/3-3-1.html The concept of a stream cipher is described in detail at  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/2-1-5 http://www.rsasecurity.com/rsalabs/faq/2-1-5 8.3 Exponentiation Ciphers You can find some information about exponentiation ciphers and many related topics in a special publication from NIST about public-key cryptography at  HYPERLINK http://csrc.nist.gov/nistpubs/800-2.txt http://csrc.nist.gov/nistpubs/800-2.txt 8.4 Public-Key Cryptography Page 285 A description of public-key cryptography, private-key cryptography, and the advantages and disadvantages of each can be found in the RSA Laboratories Cryptography FAQ at  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/2-1-1 http://www.rsasecurity.com/rsalabs/faq/2-1-1  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/2-1-2 http://www.rsasecurity.com/rsalabs/faq/2-1-2  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/2-1-3 http://www.rsasecurity.com/rsalabs/faq/2-1-3 Page 286 Information about the RSA Cryptosystem can be found at the RSA Laboratories Cryptography FAQ at  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/3-1-1.html http://www.rsasecurity.com/rsalabs/faq/3-1-1.html You can find Ronald L. Rivest's home page which contains a photograph and links to many sites related to cryptography at  HYPERLINK http://theory.lcs.mit.edu/~rivest/ http://theory.lcs.mit.edu/~rivest/ Adi Shamir's home page at the Weizmann Institute, currently containing only basic contact information, can be accessed at  HYPERLINK http://www.wisdom.weizmann.ac.il/people/~shamir http://www.wisdom.weizmann.ac.il/people/~shamir You can find Leonard Adleman's home page which contains a photograph, links to his papers, and a commentary on his involvement in the movie Sneakers at  HYPERLINK http://www-scf.usc.edu/~pwkr/adleman-home.html http://www-scf.usc.edu/~pwkr/adleman-home.html The RSA public key cryptosystem is implemented in C++ as part of the Crypto++ Library which is accessible at  HYPERLINK http://www.eskimo.com/~weidai/cryptlib http://www.eskimo.com/~weidai/cryptlib 8.5 Knapsack Ciphers Page 293 A discussion of knapsack ciphers (which are special cases of lattice-based cryptosystems) can be found at  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/2-3-11.html http://www.rsasecurity.com/rsalabs/faq/2-3-11.html 8.6 Cryptographic Protocols and Applications Page 299 You can learn more about the Diffie-Hellman scheme for key agreement at  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/3-6-1.html http://www.rsasecurity.com/rsalabs/faq/3-6-1.html The Diffie-Hellman key agreement scheme is implemented in C++ as part of the Crypto++ Library which is accessible at  HYPERLINK http://www.eskimo.com/~weidai/cryptlib.html http://www.eskimo.com/~weidai/cryptlib.html Page 300 The concept of a digital signature is explained at  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/2-2-2.html http://www.rsasecurity.com/rsalabs/faq/2-2-2.html Page 303 The basic concepts of secret sharing are described at  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/3-2-1.html http://www.rsasecurity.com/rsalabs/faq/3-2-1.html You can learn about some particular secret sharing schemes at  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/3-6-12.html http://www.rsasecurity.com/rsalabs/faq/3-6-12.html Shamir's secret sharing scheme implemented in C++ is available as part of the Crytpo++ Library which is accessible at  HYPERLINK http://www.eskimo.com/~weidai/cryptlib.html http://www.eskimo.com/~weidai/cryptlib.html 9.1 The Order of an Integer and Primitive Roots Page 308 Source code for a C program that computes the order of an integer with respect to an integer modulus can be found at  HYPERLINK http://www.mindspring.com/~pate/course/chap01.html http://www.mindspring.com/~pate/course/chap01.html 9.2 Primitive Roots for Primes Page 317 Data concerning the least primitive root of primes not exceeding 8910000000000 have been calculated by Toms Oliveira e Silva and are accessible at  HYPERLINK http://www.inesca.pt/~tos/p-roots.html http://www.inesca.pt/~tos/p-roots.html 9.3 The Existence of Primitive Roots Page 327 Source code for a program that finds a primitive root of an integer when one exists can be found at  HYPERLINK http://www.mindspring.com/~pate/course/chap01.html http://www.mindspring.com/~pate/course/chap01.html 9.4 Index Arithmetic Page 332 A discussion of the discrete logarithm problem can be found as part of the RSA Labs FAQ at  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/2-3-7.html http://www.rsasecurity.com/rsalabs/faq/2-3-7.html 9.5 Primality Tests Using Orders of Integers and Primitive Roots You can download software that runs on PCs for running Proth's primality test, and check out the latest discoveries made using this test at  HYPERLINK http://vamri.xray.ufl.edu/proths/ http://vamri.xray.ufl.edu/proths/ Page 345 To find out more about Sierpinski numbers and the quest to show that 78,557 is the smallest Sierpinski number, see  HYPERLINK http://vamri.xray.ufl.edu/proths/sierp.HTML http://vamri.xray.ufl.edu/proths/sierp.HTML 9.6 Universal Exponents Information about minimal universal exponents, which are the same as the values of the Carmichael function, can be found at Eric Weisstein's World of Mathematics at  HYPERLINK http://mathworld.wolfram.com/CarmichaelFunction.html http://mathworld.wolfram.com/CarmichaelFunction.html 10.1 Pseudorandom Numbers A useful exposition about random number generators can be found in the RSA Laboratories Cryptography FAQ at  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/2-5-2 http://www.rsasecurity.com/rsalabs/faq/2-5-2 A useful resource for the study of the generation of random numbers is the pLab Server on the Theory and Practice of Random Number Generation at  HYPERLINK http://random.mat.sbg.ac.at/ http://random.mat.sbg.ac.at/ 10.2 The ElGamal Cryptosystem Page 367 You can find the Federal Information Processing Standard (FIPS) 186 on the NIST Web site at  HYPERLINK http://csrc.nist.gov/fips/fips1861.pdf http://csrc.nist.gov/fips/fips1861.pdf A C++ implementation of the ElGamal cryptosystem is included in the Crypto++ Library at  HYPERLINK http://www.eskimo.com/~weidai/crytlib.html http://www.eskimo.com/~weidai/crytlib.html 10.3 An Application to the Splicing of Telephone Cables Page 371 To learn about splicing of coaxial cables, you may want to consult  HYPERLINK http://compnetworking.about.com/compute/compnetworking/library/weekly/aa042098.htm?terms=coaxial http://compnetworking.about.com/compute/compnetworking/library/weekly/aa042098.htm?terms=coaxial 11.2 The Law of Quadratic Reciprocity Page 392 Close to 200 different proofs of the law of quadratic reciprocity are listed by Franz Lemmermeyer. He lists the author, year, and method of each proof, and provides references and links to reviews in Mathematical Reviews and Zentralblatt. Go to  HYPERLINK http://www.rzuser.uni-heidelberg.de/~hb3/fchrono.html http://www.rzuser.uni-heidelberg.de/~hb3/fchrono.html An excellent account of Eisenstein's simplification of Gauss's third proof of the law of quadratic reciprocity can be found in "Gau, Eisenstein, and the "third" proof of the Quadratic Reciprocity Theorem: Ein kleines Schauspiel" by Reihard Laubenbacher - David J. Pengelley. This article originally appeared in the Mathematical Intelligencer in 1994 and can be accessed at  HYPERLINK http://www.math.nmsu.edu/~history/schauspiel/schauspiel.html http://www.math.nmsu.edu/~history/schauspiel/schauspiel.html 11.3 The Jacobi Symbol Page 404 You can find source code for a C program that computes Jacobi symbols at  HYPERLINK http://www.mindspring.com/~pate/crypto/chap02.html http://www.mindspring.com/~pate/crypto/chap02.html 11.4 Euler Pseudoprimes Page 413 Additional information about Euler pseudoprimes and related types of pseudoprimes can be found at  HYPERLINK http://mathworld.wolfram.com/Euler-JacobiPseudoprime.html http://mathworld.wolfram.com/Euler-JacobiPseudoprime.html 11.5 Zero-Knowledge Proofs Page 421 You can learn more about zero-knowledge proofs and interactive proofs at  HYPERLINK http://www.rsasecurity.com/rsalabs/faq/2-1-8.html http://www.rsasecurity.com/rsalabs/faq/2-1-8.html 12.1 Decimal Fractions Page 431 You can find programs that you can run online for converting fractions to decimals and decimals to fractions at  HYPERLINK http://school.discovery.com/homeworkhelp/webmath/prealgebra.html http://school.discovery.com/homeworkhelp/webmath/prealgebra.html Page 434 You can find an excellent exposition written by Helmut Richter concerning the period length of decimal fractions at  HYPERLINK http://www.lrz-muenchen.de/~hr/numb/period.html http://www.lrz-muenchen.de/~hr/numb/period.html 12.2 Finite Continued Fractions Page 443 You can learn more about continued fractions at the site  HYPERLINK http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cflINTRO.html http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cflINTRO.html 12.3 Infinite Continued Fractions Page 458 The best rational approximations of real numbers are discussed at  HYPERLINK http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html#convergents http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html#convergents 12.4 Periodic Continued Fractions Page 463 Continued fractions for square-roots are discussed at  HYPERLINK http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html#sqrtcf http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html#sqrtcf 12.5 Factoring Using Continued Fractions Page 477 An implementation of factoring using continued fractions can be found at  HYPERLINK http://www.mindspring.com/~pate/riesel/chap06.html http://www.mindspring.com/~pate/riesel/chap06.html 13.2 Fermat's Last Theorem Page 488 An excellent survey of the history of Fermat's last theorem can be found at  HYPERLINK http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html Information about Fermat's last theorem can be found at NOVA Web site on the pages accompanying their episode devoted to Wiles's proof :  HYPERLINK http://www.pbs.org/wgbh/nova/proof http://www.pbs.org/wgbh/nova/proof To begin exploring the mathematics behind Wiles's proof of Fermat's last theorem, you should look at a page developed by Charles Daney at  HYPERLINK http://www.best.com/~cgd/home/flt/flt01.htm http://www.best.com/~cgd/home/flt/flt01.htm Page 491 You can learn more about the Wolfskehl prize by downloading the article by Barner from the Notices of the American Mathematical Society at  HYPERLINK http://www.ams.org/notices/199710/barner.pdf http://www.ams.org/notices/199710/barner.pdf or by reading an account of the award ceremony and the history of the prize written by Simon Singh the author of the best selling book Fermat's Last Theorem at  HYPERLINK http://www.telegraph.co.uk/et?ac=000111151604876&rmto=33c6aeae@atmo=33c6aeae&pg=/et/97/7/5/esferm05.html http://www.telegraph.co.uk/et?ac=000111151604876&rmto=33c6aeae@atmo=33c6aeae&pg=/et/97/7/5/esferm05.html (At last, Fermat can rest in peace) 13.3 Sums of Squares Page 502 Information about Waring's problem can be found on the following pages:  HYPERLINK http://www.mathsoft.com/asolve/pwrs32/waring.html http://www.mathsoft.com/asolve/pwrs32/waring.html  HYPERLINK http://www.mathsoft.com/asolve/pwrs32/pwrs32.html http://www.mathsoft.com/asolve/pwrs32/pwrs32.html 13.4 Pell's Equation Page 507 An interactive program for solving Pell's equation can be found at the site  HYPERLINK http://www.ibc.wustl.edu/~zuker/cgi-bin/dq.html http://www.ibc.wustl.edu/~zuker/cgi-bin/dq.html Appendix A Axioms for the Set of Integers Page 515 You can learn more about the Peano axioms at  HYPERLINK http://www.torget.se/users/m/mauritz/math/num/peano.htm http://www.torget.se/users/m/mauritz/math/num/peano.htm Appendix C Using Maple and Mathematica for Number Theory Page 527 The Maple home page is a good place to starting learning more about Maple:  HYPERLINK http://www.maplesoft.com http://www.maplesoft.com Page 528 Maple worksheets written by John Cosgrave for a course in number theory and cryptography at St. Patrick's College in Dublin, Ireland can be found at  HYPERLINK http://www.spd.dcu.ie/johnbcos/Maple_3rd_year.htm http://www.spd.dcu.ie/johnbcos/Maple_3rd_year.htm You can find a Maple worksheet on the Collatz problem at  HYPERLINK http://www.maplesoft.com/apps/categories/mathematics/numbertheory/html/collatz.html http://www.maplesoft.com/apps/categories/mathematics/numbertheory/html/collatz.html Page 530 Information on Mathematica can be found at  HYPERLINK http://www.mathematica.com http://www.mathematica.com Mathematica packages can be accessed at  HYPERLINK http://www.mathsource.com http://www.mathsource.com Page 531 A program for implementing the iterations in the Collatz 3x+1 problem in Mathematica can be found at  HYPERLINK http://www.mathsource.com/Content/Applications/Mathematics/0200-305 http://www.mathsource.com/Content/Applications/Mathematics/0200-305 Page 532 You can find an implementation of the RSA Public-Key Cryptosystem in Mathematica at  HYPERLINK http://www.mathsource.com/Content/Applications/ComputerScience/0204-130 http://www.mathsource.com/Content/Applications/ComputerScience/0204-130 Appendix E TABLES Page 537 You can find a table of the first 100,008 primes on the Prime Pages at  HYPERLINK http://www.utm.edu/research/primes/lists/small/100000.txt http://www.utm.edu/research/primes/lists/small/100000.txt Fe  UVW !   - 8 9 : | }  : ; F G   K jCJUj9CJUjCJUjCJUjCJUj9CJU0JCJjCJU jCJU5CJCJ@Fdem "#  - . 7 ~  $ & FFdem "#  - . 7 ~  : F  r s YJS)wDh2YZvwPQUVU_JTLTs b : F  r s YJS)$K L M p q 345~STU%&'WXHI'(wxXYhi5CJj&CJUj CJUjCJUjkCJUj2 CJUjy CJUj CJUCJ0JCJ jCJUjCJU:)wDh2YZvwPQUVU$23jklWXYuNOQ !"STSTHIJKjCJUj}CJUj CJUjCJUjCJU5CJj!CJUjCJUCJ0JCJ jCJUjCJU<<<@=J=z===(>>>>>G??????*@z@@@NAPAAAB$BBB&CCCC DDDD(EEEFFFFFFF GGe111112222222t3u33333333445464`4a444444 5 5H5I55555555B6C6y6z6{6665767i7j7k777778j?BCJUjACJU5CJj?CJUj>CJUj=CJUj<CJUj;CJUCJ0JCJ jCJUj:CJU:111P2223t3333b44 5H555555'6B666757777Q8$88 8O8P88888899 9(9q9r999999C:D::::::@;A;;;;;;><?<w<x<y<<<<<===>=?=z={======(>)>a>jKCJUjJCJUjICJUj8HCJUjFCJUjECJU5CJjDCJUCJ0JCJ jCJUjTCCJU:Q8R889 9'9(929q999/::;,;;;;(<><<<@=J=z===(>>>$a>b>c>>>>>G?H?{?|?}?????z@{@@@@@@@@AA ALAMAPAQAAAAAA8B9BvBwBxBBB:C;CiCjCkCCCD D[D\D]DDD>>G??????*@z@@@NAPAAAB$BBB&CCCC DDDD(E$wExEyEEE-F.F`FaFbFFFFF,G-GkGlGmGGGGG$H%HbHcHdHHHIIGIHIIItIuIIIIII%J&JgJhJJJJJJKKYKZK[KKKLL?Lj^CJUjf]CJUj)\CJUj[CJUjYCJUjXCJU5CJjuWCJUCJ0JCJ jCJUjDVCJU<(EEEFFFFFFF GGGGGGG$HHHHIvIII'J1JgJJJ$GGGGGG$HHHHIvIII'J1JgJJJKKKKLnLoLLLLM!MMMMMMNWNNNNNNKOOOOOO>PPPPPRQQQQQP?P|P}P~PPPPPQQQQQQQTRURRRRRRRR|S}SSjfCJUjeCJUjdCJUj[cCJUjJbCJUj aCJU5CJCJ0JCJ jCJUj_CJU=OO>PPPPPRQQQQQfHffHgIggg(h*hhi&iiijjkkkkk3lllmm1mEmOmm nn7nAnnnnn(o2o}oooo%pd\\\#]$]b]c]d]]]]] ^!^f^g^h^^^^^__T_U_V_____2`3``````BaCa~aaaaaaa+b,b|b}b~bbbbbJcKcccccccdsdtddjIzCJUjxCJUjwCJUjvCJUjtCJUjxsCJU0JCJj7rCJU jCJU5CJCJB\]]]]]] ^^^^^^^______!`2```,aBaaaaa$aa+bbbbbcJcccd2dfHffHgIgg$dddeeeDeeeeee f f f'ffffffFgGggghhh&h'hhhhhhiiiiiiijjjj2k3k4kkkkk3l4lqlrlslllllj CJUjCJUj˂CJUjCJUjCJUj~CJUj}CJU5CJCJ0JCJ jCJUj|CJU=gg(h*hhi&iiijjkkkkk3lllmm1mEmOmm nn7nAn$llllmmm1mmmmmm n nnnonnnnnnn}o~oooooo_p`ppppppqq{q|q}qqq r r0r1r2rLrMrwrxrrrrrr>s?ssjяCJUjCJUj)CJUjCJUjCJUjCJUjCJU5CJ0JCJ jCJUjJCJUCJhttp://www.maths.usyd.edu.au:8000/u/magma/Examples/node6.htmlyK |http://www.maths.usyd.edu.au:8000/u/magma/Examples/node6.htmlADyK 3http://www.mindspring.com/~pate/crypto/chap02.htmlyK fhttp://www.mindspring.com/~pate/crypto/chap02.html=DyK 2http://www.rsasecurity.com/rsalabs/faq/2-3-4.htmlyK dhttp://www.rsasecurity.com/rsalabs/faq/2-3-4.htmlqDyK ?http://mathworld.wolfram.com/PrimeFactorizationAlgorithms.htmlyK ~http://mathworld.wolfram.com/PrimeFactorizationAlgorithms.htmlYDyK 9http://www.rsasecurity.com/rsalabs/challenges/factoring/yK rhttp://www.rsasecurity.com/rsalabs/challenges/factoring/)DyK -http://vamri.xray.ufl.edu/proths/fermat.htmlyK Zhttp://vamri.xray.ufl.edu/proths/fermat.htmlDyK #http://www.perfsci.com/prizes.htmlyK Fhttp://www.perfsci.com/prizes.html%DyK ,http://www.cerias.purdue.edu/homes/ssw/cun/yK Xhttp://www.cerias.purdue.edu/homes/ssw/cun/EDyK 4http://www.thoralf2.uwaterloo.ca/htdocs/linear.htmlyK hhttp://www.thoralf2.uwaterloo.ca/htdocs/linear.html)DyK -http://www.cut-the-knot.com/blue/Modulo.htmlyK Zhttp://www.cut-the-knot.com/blue/Modulo.htmlADyK 3http://www.mindspring.com/~pate/crypto/chap02.htmlyK fhttp://www.mindspring.com/~pate/crypto/chap02.htmlEDyK 4http://www.unc.edu/~rowlett/Math81/notes/Oct06.htmlyK hhttp://www.unc.edu/~rowlett/Math81/notes/Oct06.htmlADyK 3http://www.mindspring.com/~pate/crypto/chap02.htmlyK fhttp://www.mindspring.com/~pate/crypto/chap02.html-DyK .http://www.cut-the-knot.com/blue/chinese.htmlyK \http://www.cut-the-knot.com/blue/chinese.htmlDyK  http://www.facade.com/biorhythmyK @http://www.facade.com/biorhythmUDyK 8http://forum.swarthmore.edu/alejandre/magic.square.htmlyK phttp://forum.swarthmore.edu/alejandre/magic.square.htmlADyK 3http://www.mindspring.com/~pate/course/chap08.htmlyK fhttp://www.mindspring.com/~pate/course/chap08.htmlADyK 3http://forum.swarthmore.edu/k12/mathtips/ward.htmlyK fhttp://forum.swarthmore.edu/k12/mathtips/ward.htmlYDyK 9http://www.utm.edu/research/primes/glossary/Repunit.htmlyK rhttp://www.utm.edu/research/primes/glossary/Repunit.html)DyK -http://www.execpc.com/~mikeber/calendar.htmlyK Zhttp://www.execpc.com/~mikeber/calendar.htmlYDyK 9http://www.comp.nus.edu.sg/~henz/projects/acc/index.htmlyK rhttp://www.comp.nus.edu.sg/~henz/projects/acc/index.htmlDyK Chttp://www.whatis.com/WhatIs_Definition_Page/0,4152,212230,00.htmlyK http://www.whatis.com/WhatIs_Definition_Page/0,4152,212230,00.html=DyK 2http://www.rsasecurity.com/rsalabs/faq/2-1-6.htmlyK dhttp://www.rsasecurity.com/rsalabs/faq/2-1-6.htmliDyK =http://www.augustana.ab.ca/~mohrj/algorithms/checkdigit.htmlyK zhttp://www.augustana.ab.ca/~mohrj/algorithms/checkdigit.html)DyK ,http://www.albar.co.uk/tooltips.htm#isbncdcyK Hhttp://www.albar.co.uk/tooltips.htmisbncdc)DyK ,http://www.albar.co.uk/tooltips.htm#upcacdcyK Hhttp://www.albar.co.uk/tooltips.htmupcacdcDyK Lhttp://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Perfect_numbers.htmlyK http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Perfect_numbers.html=DyK 2http://www.utm.edu/research/primes/mersenne.shtmlyK dhttp://www.utm.edu/research/primes/mersenne.shtml=DyK 2http://www.utm.edu/research/primes/mersenne.shtmlyK dhttp://www.utm.edu/research/primes/mersenne.shtml!DyK +http://www.scruznet.com/~luke/mersenne.htmyK Vhttp://www.scruznet.com/~luke/mersenne.htmDyK http://www.mersenneyK *http://www.mersenne/DyK http://www.entropia.com/ips/yK :http://www.entropia.com/ips/!DyK +http://xraysgi.ims.uconn.edu/amicable.htmlyK Vhttp://xraysgi.ims.uconn.edu/amicable.htmlADyK 3http://www.vejlehs.dk/staff/jmp/aliquot/knwnap.htmyK fhttp://www.vejlehs.dk/staff/jmp/aliquot/knwnap.htm%DyK ,http://www.uni-bielefeld.de/~achim/mpn.htmlyK Xhttp://www.uni-bielefeld.de/~achim/mpn.html!DyK +http://www.faqs.org/faqs/cryptography-faq/yK Vhttp://www.faqs.org/faqs/cryptography-faq/DyK (http://www.rsasecurity.com/rsalabs/faq/yK Phttp://www.rsasecurity.com/rsalabs/faq/5DyK 0http://www.rsasecurity.com/rsalabs/faq/1-2.htmlyK `http://www.rsasecurity.com/rsalabs/faq/1-2.htmlDyK )http://www.ssh.fi/tech/crypto/intro.htmlyK Rhttp://www.ssh.fi/tech/crypto/intro.html=DyK 2http://www.rsasecurity.com/rsalabs/faq/2-1-4.htmlyK dhttp://www.rsasecurity.com/rsalabs/faq/2-1-4.htmlYDyK 9http://www.trincoll.edu/depts/cpsc/cryptography/vigenereyK rhttp://www.trincoll.edu/depts/cpsc/cryptography/vigenereQDyK 7http://www.cs.arizona.edu/people/muth/Cipher/query_vcbyK nhttp://www.cs.arizona.edu/people/muth/Cipher/query_vcb)DyK -http://www.itl.nist.gov/fipspubs/fip46-2.htmyK Zhttp://www.itl.nist.gov/fipspubs/fip46-2.htm)DyK -http://www.rsasecurity.com/rsalabs/faq/3-2-1yK Zhttp://www.rsasecurity.com/rsalabs/faq/3-2-1=DyK 2http://www.rsasecurity.com/rsalabs/faq/3-3-1.htmlyK dhttp://www.rsasecurity.com/rsalabs/faq/3-3-1.html)DyK -http://www.rsasecurity.com/rsalabs/faq/2-1-5yK Zhttp://www.rsasecurity.com/rsalabs/faq/2-1-5DyK (http://csrc.nist.gov/nistpubs/800-2.txtyK Phttp://csrc.nist.gov/nistpubs/800-2.txt)DyK -http://www.rsasecurity.com/rsalabs/faq/2-1-1yK Zhttp://www.rsasecurity.com/rsalabs/faq/2-1-1)DyK -http://www.rsasecurity.com/rsalabs/faq/2-1-2yK Zhttp://www.rsasecurity.com/rsalabs/faq/2-1-2)DyK -http://www.rsasecurity.com/rsalabs/faq/2-1-3yK Zhttp://www.rsasecurity.com/rsalabs/faq/2-1-3=DyK 2http://www.rsasecurity.com/rsalabs/faq/3-1-1.htmlyK dhttp://www.rsasecurity.com/rsalabs/faq/3-1-1.htmlDyK #http://theory.lcs.mit.edu/~rivest/yK Fhttp://theory.lcs.mit.edu/~rivest/5DyK 0http://www.wisdom.weizmann.ac.il/people/~shamiryK `http://www.wisdom.weizmann.ac.il/people/~shamir1DyK /http://www-scf.usc.edu/~pwkr/adleman-home.htmlyK ^http://www-scf.usc.edu/~pwkr/adleman-home.htmlDyK 'http://www.eskimo.com/~weidai/cryptlibyK Nhttp://www.eskimo.com/~weidai/cryptlibADyK 3http://www.rsasecurity.com/rsalabs/faq/2-3-11.htmlyK fhttp://www.rsasecurity.com/rsalabs/faq/2-3-11.html=DyK 2http://www.rsasecurity.com/rsalabs/faq/3-6-1.htmlyK dhttp://www.rsasecurity.com/rsalabs/faq/3-6-1.html%DyK ,http://www.eskimo.com/~weidai/cryptlib.htmlyK Xhttp://www.eskimo.com/~weidai/cryptlib.html=DyK 2http://www.rsasecurity.com/rsalabs/faq/2-2-2.htmlyK dhttp://www.rsasecurity.com/rsalabs/faq/2-2-2.html=DyK 2http://www.rsasecurity.com/rsalabs/faq/3-2-1.htmlyK dhttp://www.rsasecurity.com/rsalabs/faq/3-2-1.htmlADyK 3http://www.rsasecurity.com/rsalabs/faq/3-6-12.htmlyK fhttp://www.rsasecurity.com/rsalabs/faq/3-6-12.html%DyK ,http://www.eskimo.com/~weidai/cryptlib.htmlyK Xhttp://www.eskimo.com/~weidai/cryptlib.htmlADyK 3http://www.mindspring.com/~pate/course/chap01.htmlyK fhttp://www.mindspring.com/~pate/course/chap01.htmlDyK 'http://www.inesca.pt/~tos/p-roots.htmlyK Nhttp://www.inesca.pt/~tos/p-roots.htmlADyK 3http://www.mindspring.com/~pate/course/chap01.htmlyK fhttp://www.mindspring.com/~pate/course/chap01.html=DyK 2http://www.rsasecurity.com/rsalabs/faq/2-3-7.htmlyK dhttp://www.rsasecurity.com/rsalabs/faq/2-3-7.htmlDyK "http://vamri.xray.ufl.edu/proths/yK Dhttp://vamri.xray.ufl.edu/proths/%DyK ,http://vamri.xray.ufl.edu/proths/sierp.HTMLyK Xhttp://vamri.xray.ufl.edu/proths/sierp.HTMLIDyK 5http://mathworld.wolfram.com/CarmichaelFunction.htmlyK jhttp://mathworld.wolfram.com/CarmichaelFunction.html)DyK -http://www.rsasecurity.com/rsalabs/faq/2-5-2yK Zhttp://www.rsasecurity.com/rsalabs/faq/2-5-2DyK http://random.mat.sbg.ac.at/yK :http://random.mat.sbg.ac.at/DyK 'http://csrc.nist.gov/fips/fips1861.pdfyK Nhttp://csrc.nist.gov/fips/fips1861.pdf!DyK +http://www.eskimo.com/~weidai/crytlib.htmlyK Vhttp://www.eskimo.com/~weidai/crytlib.htmlDyK ahttp://compnetworking.about.com/compute/compnetworking/library/weekly/aa042098.htm?terms=coaxialyK http://compnetworking.about.com/compute/compnetworking/library/weekly/aa042098.htm?terms=coaxialMDyK 6http://www.rzuser.uni-heidelberg.de/~hb3/fchrono.htmlyK lhttp://www.rzuser.uni-heidelberg.de/~hb3/fchrono.htmliDyK =http://www.math.nmsu.edu/~history/schauspiel/schauspiel.htmlyK zhttp://www.math.nmsu.edu/~history/schauspiel/schauspiel.htmlADyK 3http://www.mindspring.com/~pate/crypto/chap02.htmlyK fhttp://www.mindspring.com/~pate/crypto/chap02.html]DyK :http://mathworld.wolfram.com/Euler-JacobiPseudoprime.htmlyK thttp://mathworld.wolfram.com/Euler-JacobiPseudoprime.html=DyK 2http://www.rsasecurity.com/rsalabs/faq/2-1-8.htmlyK dhttp://www.rsasecurity.com/rsalabs/faq/2-1-8.htmlyDyK Ahttp://school.discovery.com/homeworkhelp/webmath/prealgebra.htmlyK http://school.discovery.com/homeworkhelp/webmath/prealgebra.html5DyK 0http://www.lrz-muenchen.de/~hr/numb/period.htmlyK `http://www.lrz-muenchen.de/~hr/numb/period.htmlDyK Ehttp://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cflINTRO.htmlyK http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cflINTRO.htmlDyK Phttp://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html#convergentsyK http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html convergentsDyK Khttp://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html#sqrtcfyK http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.htmlsqrtcfADyK 3http://www.mindspring.com/~pate/riesel/chap06.htmlyK fhttp://www.mindspring.com/~pate/riesel/chap06.htmlDyK Rhttp://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.htmlyK http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.htmlDyK #http://www.pbs.org/wgbh/nova/proofyK Fhttp://www.pbs.org/wgbh/nova/proof%DyK ,http://www.best.com/~cgd/home/flt/flt01.htmyK Xhttp://www.best.com/~cgd/home/flt/flt01.htm)DyK -http://www.ams.org/notices/199710/barner.pdfyK Zhttp://www.ams.org/notices/199710/barner.pdfDyK ihttp://www.telegraph.co.uk/et?ac=000111151604876&rmto=33c6aeae@atmo=33c6aeae&pg=/et/97/7/5/esferm05.htmlyK http://www.telegraph.co.uk/et?ac=000111151604876&rmto=33c6aeae@atmo=33c6aeae&pg=/et/97/7/5/esferm05.html=DyK 2http://www.mathsoft.com/asolve/pwrs32/waring.htmlyK dhttp://www.mathsoft.com/asolve/pwrs32/waring.html=DyK 2http://www.mathsoft.com/asolve/pwrs32/pwrs32.htmlyK dhttp://www.mathsoft.com/asolve/pwrs32/pwrs32.html5DyK 0http://www.ibc.wustl.edu/~zuker/cgi-bin/dq.htmlyK `http://www.ibc.wustl.edu/~zuker/cgi-bin/dq.htmlUDyK 8http://www.torget.se/users/m/mauritz/math/num/peano.htmyK phttp://www.torget.se/users/m/mauritz/math/num/peano.htmDyK http://www.maplesoft.comyK 4http://www.maplesoft.com/=DyK 2http://www.spd.dcu.ie/johnbcos/Maple_3rd_year.htmyK dhttp://www.spd.dcu.ie/johnbcos/Maple_3rd_year.htmDyK Thttp://www.maplesoft.com/apps/categories/mathematics/numbertheory/html/collatz.htmlyK http://www.maplesoft.com/apps/categories/mathematics/numbertheory/html/collatz.htmlDyK http://www.mathematica.comyK 8http://www.mathematica.com/DyK http://www.mathsource.comyK 6http://www.mathsource.com/DyK Dhttp://www.mathsource.com/Content/Applications/Mathematics/0200-305yK http://www.mathsource.com/Content/Applications/Mathematics/0200-305DyK Hhttp://www.mathsource.com/Content/Applications/ComputerScience/0204-130yK http://www.mathsource.com/Content/Applications/ComputerScience/0204-130]DyK :http://www.utm.edu/research/primes/lists/small/100000.txtyK thttp://www.utm.edu/research/primes/lists/small/100000.txt [0@0NormalCJOJQJmH 0@0 Heading 1$@&5<A@<Default Paragraph Font(U@( Hyperlink>*B**>@*Title$5CJ ,J@,Subtitle$58V@!8FollowedHyperlink>*B* qK P" *18a>wE?LS\dlsu;?ACFHKMOQUWZ]_b )U$*1Q8>(EJO|U\agAntu<>@BEGILNPRTVX[\^`c0G\%pu=DJSYa V 9|:F L p 4 ~ T & W  H 'wXh2kWN!SSHJsV7q)iS3f!Ow !!"y"""p###$%R%% &P&&)'[''7(t(()A))) **+L+++,\,,,---...t////50`000 1H111B2z2253j3334O4445q555C666@777>8x8889>9z999(:b::G;|;;z<<<<=L=P===8>w>>:?j??@\@@L}LLMMMTNNN|OOOPPP|QQQaRRR,SdSS,TTT'VjVVXcXX#YcYY ZgZZ[U[[2\\\B]]]+^}^^J___s``aaa bbbFccd&dddeeeff3gg3hrhhhhiii jnjjj}kkk_lllm|mm n1nLnwnnn>oooJpppcqqqqXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX _Hlt531594685\q\q$,-4$NW x 'sy/:  J M 5"="M"U"R&S&))$*,*v+~+1,9,,-N-S-T-\-p-x-.#.T.\.../ /./6/(2,23 3666666 77==>>??????@@AAAABCCDDDDDGGGG)J.J/J7JLL,M3M4M=MMN ???@@@@BB C CIIJ%JRMUMT,TUU!\#\,]3]ccffqMichelle DeDeo+C:\~mdedeo\Math-Work Related\numbthyweb.docMichelle DeDeo+C:\~mdedeo\Math-Work Related\numbthyweb.docMichelle DeDeo3C:\windows\TEMP\AutoRecovery save of numbthyweb.asdMichelle DeDeo+C:\~mdedeo\Math-Work Related\numbthyweb.docMichelle DeDeo+C:\~mdedeo\Math-Work Related\numbthyweb.docMichelle DeDeo+C:\~mdedeo\Math-Work Related\numbthyweb.docMichelle DeDeo3C:\windows\TEMP\AutoRecovery save of numbthyweb.asdMichelle DeDeo3C:\windows\TEMP\AutoRecovery save of numbthyweb.asdMichelle DeDeo3C:\windows\TEMP\AutoRecovery save of numbthyweb.asdMichelle DeDeo+C:\~mdedeo\Math-Work Related\numbthyweb.docmKPzo(zo(.0o(..88o(... 88o( .... `o( ..... `o( ...... o(....... o(........m@R&R&2R&R&q@@GTimes New Roman5Symbol3& Arial"qhs[f[f3y]0i20Usd1Michelle DeDeoMichelle DeDeo Oh+'0x  4 @ LX`hp1ssMichelle DeDeoichich Normal.dotDMichelle DeDeo14hMicrosoft Word 8.0@@B=x@OKxy] ՜.+,D՜.+,D hp  University of North Florida0Us1 1 TitleH(RZ _PID_GUID _PID_HLINKSAN{6A3DBD55-E408-11D5-A985-00C04F2718D7}AHPb:http://www.utm.edu/research/primes/lists/small/100000.txtfe_Hhttp://www.mathsource.com/Content/Applications/ComputerScience/0204-130on\Dhttp://www.mathsource.com/Content/Applications/Mathematics/0200-305/9Yhttp://www.mathsource.com/b0Vhttp://www.mathematica.com/--SThttp://www.maplesoft.com/apps/categories/mathematics/numbertheory/html/collatz.htmlP2http://www.spd.dcu.ie/johnbcos/Maple_3rd_year.htm FMhttp://www.maplesoft.com/%;J8http://www.torget.se/users/m/mauritz/math/num/peano.htm#`G0http://www.ibc.wustl.edu/~zuker/cgi-bin/dq.htmlFD2http://www.mathsoft.com/asolve/pwrs32/pwrs32.htmlGA2http://www.mathsoft.com/asolve/pwrs32/waring.htmli>ihttp://www.telegraph.co.uk/et?ac=000111151604876&rmto=33c6aeae@atmo=33c6aeae&pg=/et/97/7/5/esferm05.htmlF;-http://www.ams.org/notices/199710/barner.pdf*<8,http://www.best.com/~cgd/home/flt/flt01.htmdm5#http://www.pbs.org/wgbh/nova/proof> 2Rhttp://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html!p/3http://www.mindspring.com/~pate/riesel/chap06.htmlG,Dhttp://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.htmlsqrtcf^)Dhttp://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html convergentsE&Ehttp://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cflINTRO.html>5#0http://www.lrz-muenchen.de/~hr/numb/period.html N Ahttp://school.discovery.com/homeworkhelp/webmath/prealgebra.html2http://www.rsasecurity.com/rsalabs/faq/2-1-8.html]A:http://mathworld.wolfram.com/Euler-JacobiPseudoprime.html9k3http://www.mindspring.com/~pate/crypto/chap02.html =http://www.math.nmsu.edu/~history/schauspiel/schauspiel.htmlP6http://www.rzuser.uni-heidelberg.de/~hb3/fchrono.htmlZahttp://compnetworking.about.com/compute/compnetworking/library/weekly/aa042098.htm?terms=coaxial,/ +http://www.eskimo.com/~weidai/crytlib.html+$'http://csrc.nist.gov/fips/fips1861.pdfWXhttp://random.mat.sbg.ac.at/G-http://www.rsasecurity.com/rsalabs/faq/2-5-2] 5http://mathworld.wolfram.com/CarmichaelFunction.html/{,http://vamri.xray.ufl.edu/proths/sierp.HTMLA"http://vamri.xray.ufl.edu/proths/2http://www.rsasecurity.com/rsalabs/faq/2-3-7.html1~3http://www.mindspring.com/~pate/course/chap01.htmlo 'http://www.inesca.pt/~tos/p-roots.html1~3http://www.mindspring.com/~pate/course/chap01.html`,http://www.eskimo.com/~weidai/cryptlib.htmlp"3http://www.rsasecurity.com/rsalabs/faq/3-6-12.html2http://www.rsasecurity.com/rsalabs/faq/3-2-1.html2http://www.rsasecurity.com/rsalabs/faq/2-2-2.html`,http://www.eskimo.com/~weidai/cryptlib.html2http://www.rsasecurity.com/rsalabs/faq/3-6-1.htmls&3http://www.rsasecurity.com/rsalabs/faq/2-3-11.html:z'http://www.eskimo.com/~weidai/cryptlib~~/http://www-scf.usc.edu/~pwkr/adleman-home.html-z0http://www.wisdom.weizmann.ac.il/people/~shamirg#http://theory.lcs.mit.edu/~rivest/2http://www.rsasecurity.com/rsalabs/faq/3-1-1.htmlG-http://www.rsasecurity.com/rsalabs/faq/2-1-3G-http://www.rsasecurity.com/rsalabs/faq/2-1-2G-http://www.rsasecurity.com/rsalabs/faq/2-1-1?s(http://csrc.nist.gov/nistpubs/800-2.txtG-http://www.rsasecurity.com/rsalabs/faq/2-1-52http://www.rsasecurity.com/rsalabs/faq/3-3-1.htmlG-http://www.rsasecurity.com/rsalabs/faq/3-2-1V-http://www.itl.nist.gov/fipspubs/fip46-2.htm)7http://www.cs.arizona.edu/people/muth/Cipher/query_vcbR9http://www.trincoll.edu/depts/cpsc/cryptography/vigenere2http://www.rsasecurity.com/rsalabs/faq/2-1-4.htmlF)http://www.ssh.fi/tech/crypto/intro.html0&0http://www.rsasecurity.com/rsalabs/faq/1-2.htmlh (http://www.rsasecurity.com/rsalabs/faq/s>+http://www.faqs.org/faqs/cryptography-faq/&},http://www.uni-bielefeld.de/~achim/mpn.html4=3http://www.vejlehs.dk/staff/jmp/aliquot/knwnap.htmle+http://xraysgi.ims.uconn.edu/amicable.htmlI http://www.entropia.com/ips/Mhttp://www.mersenne/rk+http://www.scruznet.com/~luke/mersenne.htm\2http://www.utm.edu/research/primes/mersenne.shtml\2http://www.utm.edu/research/primes/mersenne.shtmlI0Lhttp://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Perfect_numbers.htmlBL~$http://www.albar.co.uk/tooltips.htmupcacdc_@{$http://www.albar.co.uk/tooltips.htmisbncdcx=http://www.augustana.ab.ca/~mohrj/algorithms/checkdigit.htmlu2http://www.rsasecurity.com/rsalabs/faq/2-1-6.htmlNQrChttp://www.whatis.com/WhatIs_Definition_Page/0,4152,212230,00.htmlMLo9http://www.comp.nus.edu.sg/~henz/projects/acc/index.html l-http://www.execpc.com/~mikeber/calendar.html i9http://www.utm.edu/research/primes/glossary/Repunit.html!pf3http://forum.swarthmore.edu/k12/mathtips/ward.html8~c3http://www.mindspring.com/~pate/course/chap08.htmli-`8http://forum.swarthmore.edu/alejandre/magic.square.html*.] http://www.facade.com/biorhythmNAZ.http://www.cut-the-knot.com/blue/chinese.html9kW3http://www.mindspring.com/~pate/crypto/chap02.htmlmrT4http://www.unc.edu/~rowlett/Math81/notes/Oct06.html9kQ3http://www.mindspring.com/~pate/crypto/chap02.htmlN-http://www.cut-the-knot.com/blue/Modulo.html{sK4http://www.thoralf2.uwaterloo.ca/htdocs/linear.html1nH,http://www.cerias.purdue.edu/homes/ssw/cun/ wE#http://www.perfsci.com/prizes.html[B-http://vamri.xray.ufl.edu/proths/fermat.html[?9http://www.rsasecurity.com/rsalabs/challenges/factoring/%x<?http://mathworld.wolfram.com/PrimeFactorizationAlgorithms.html92http://www.rsasecurity.com/rsalabs/faq/2-3-4.html9k63http://www.mindspring.com/~pate/crypto/chap02.html[D3>http://www.maths.usyd.edu.au:8000/u/magma/Examples/node6.html?&0,http://www.cut-the-knot.com/blue/Farey.html -5http://www.utm.edu/research/primes/glossary/GCD.htmlTQ*Fhttp://www.informatik.uni-giessen.de/staff/richstein/ca/Goldbach.htmlC')http://www.ltkz.demon.co.uk/ktuplets.htm[$5http://www.utm.edu/research/primes/notes/conjecturesL!9http://www.utm.edu/research/primes/programs/Eratosthenes1http://www.math.utah.edu/~alfeld/math/prime.htmlFQhttp://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Eratosthenes.html6_Jhttp://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Prime_numbers.html'c$http://www.utm.edu/research/primes/H2http://mathworld.wolfram.com/KaprekarRoutine.html,{;http://www.cecm.sfu.ca/organics/papers/lagarias/index.html"c Chttp://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html_5 +http://www.cs.wm.edu/~pkstoc/h_papers.htmlG&http://www.cs.wm.edu/~pkstoc/toh.html157http://www.research.att.com/~njas/sequences/index.htmlS1http://mathworld.wolfram.com/FigurateNumber.html  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdfghijklmnopqrstuvwxyz{|}~Root Entry FK@x Ҟ[KxData e'1TableWordDocumentSummaryInformation(DocumentSummaryInformation8JCompObjjObjectPool Ҟ[Kx Ҟ[Kx  FMicrosoft Word Document MSWordDocWord.Document.89q