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CHAPTER 9 - Primitive Roots
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9.1 The Order of an Integer and Primitive Roots

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You can find the order of an integer modulo a given positive integer using the tool at http://www.numbertheory.org/php/order.html (Finding the order of a (mod m))

Source code for a C program that computes the order of an integer with respect to an integer modulus can be found at
http://www.mindspring.com/~pate/course/chap01.html

 

9.2 Primitive Roots for Primes

Page 343

You can use the applet at http://www.numbertheory.org/php/lprimroot.html to find the least primitive root of primes with fewer than 20 digits.

Data concerning the least primitive root of primes not exceeding 8910000000000 have been calculated by Tomás Oliveira e Silva and are accessible at
http://www.inesca.pt/~tos/p-roots.html (Least primitive root of prime numbers)

Biographical information about Emil Artin can be found at the MacTutor History of Mathematics Archive at
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Artin.html (Artin)

 

9.3 The Existence of Primitive Roots

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Source code for a program that finds a primitive root of an integer when one exists can be found at
http://www.mindspring.com/~pate/course/chap01.html

 

9.4 Index Arithmetic

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A discussion of the discrete logarithm problem can be found as part of the RSA Labs FAQ at
http://www.rsasecurity.com/rsalabs/faq/2-3-7.html (RSA Labs FAQ - What is the discrete logarithm problem?)

 

An applet for computing discrete logarithms can be found at http://www.alpertron.com.ar/DILOG.HTM (discrete logarithm calculator)

 

You can compute discrete logarithms using the applet at http://www.numbertheory.org/php/discrete_log.html (Shanks baby-steps/giant-steps algorithm for finding the discrete log)

 

 

9.5 Primality Tests Using Orders of Integers and Primitive Roots

 

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John Cosgrove works through Pocklington’s original paper on primality testing at http://www.spd.dcu.ie/johnbcos/download/3rd_year/Pocklington_1914/1914_PAP1.html (Thinking through Pocklington's 1914-16 paper)

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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 http://www.prothsearch.net/
 (Yves Gallot's Proth Search Page)

 

You can run a Proth’s primality test using the tool at http://www.numbertheory.org/php/proth.html (Proth’s algorithm)

 

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To find out more about Sierpinski numbers and the quest to show that 78,557 is the smallest Sierpinski number, see http://www.prothsearch.net/sierp.html

(Sierpinski problem)

 

Seventeen or bust is a distributed attack on the Sierpinski problem: http://www.seventeenorbust.com/ (Seventeen or Bust)

 

 

9.6 Universal Exponents

 

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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
http://mathworld.wolfram.com/CarmichaelFunction.html (Carmichael Function)

 

An applet for computing minimal universal exponents can be found at http://www.math-it.de/Mathematik/Zahlentheorie/Zahl/ZahlApplet.html