### Meta

### Categories

- algebra (46)
- arithmetic (83)
- books (30)
- calculus (7)
- challenges (57)
- combinatorics (28)
- complex numbers (6)
- computation (80)
- convergence (9)
- counting (36)
- famous numbers (48)
- fibonacci (18)
- fractals (13)
- games (34)
- geometry (71)
- golden ratio (8)
- group theory (28)
- humor (7)
- induction (8)
- infinity (19)
- iteration (24)
- links (76)
- logic (9)
- meta (43)
- modular arithmetic (30)
- number theory (108)
- open problems (11)
- paradox (1)
- pascal's triangle (8)
- pattern (103)
- people (21)
- pictures (71)
- posts without words (34)
- primes (55)
- probability (9)
- programming (20)
- proof (89)
- puzzles (18)
- recursion (16)
- review (21)
- sequences (28)
- solutions (31)
- teaching (15)
- trig (3)
- Uncategorized (7)
- video (19)

### Archives

- October 2019 (1)
- September 2019 (7)
- August 2019 (3)
- July 2019 (5)
- May 2019 (4)
- April 2019 (2)
- March 2019 (3)
- February 2019 (3)
- January 2019 (4)
- November 2018 (3)
- October 2018 (4)
- September 2018 (4)
- August 2018 (6)
- July 2018 (2)
- June 2018 (5)
- May 2018 (3)
- April 2018 (5)
- March 2018 (4)
- February 2018 (3)
- January 2018 (4)
- December 2017 (3)
- November 2017 (3)
- October 2017 (1)
- September 2017 (1)
- July 2017 (4)
- June 2017 (4)
- May 2017 (9)
- April 2017 (7)
- March 2017 (5)
- February 2017 (4)
- January 2017 (3)
- December 2016 (4)
- November 2016 (6)
- October 2016 (6)
- September 2016 (2)
- August 2016 (5)
- July 2016 (2)
- June 2016 (4)
- May 2016 (4)
- April 2016 (2)
- March 2016 (3)
- February 2016 (9)
- January 2016 (8)
- December 2015 (5)
- November 2015 (29)
- August 2015 (3)
- June 2015 (2)
- April 2015 (1)
- May 2014 (1)
- December 2013 (1)
- October 2013 (1)
- July 2013 (1)
- June 2013 (1)
- May 2013 (1)
- April 2013 (3)
- March 2013 (3)
- February 2013 (2)
- January 2013 (5)
- December 2012 (3)
- November 2012 (4)
- October 2012 (5)
- September 2012 (1)
- August 2012 (4)
- July 2012 (1)
- June 2012 (6)
- May 2012 (2)
- April 2012 (3)
- March 2012 (1)
- February 2012 (4)
- January 2012 (5)
- December 2011 (1)
- November 2011 (7)
- October 2011 (4)
- September 2011 (6)
- July 2011 (2)
- June 2011 (4)
- May 2011 (5)
- April 2011 (2)
- March 2011 (4)
- February 2011 (1)
- January 2011 (1)
- December 2010 (1)
- November 2010 (4)
- October 2010 (2)
- September 2010 (1)
- August 2010 (1)
- July 2010 (1)
- June 2010 (2)
- May 2010 (3)
- April 2010 (1)
- February 2010 (6)
- January 2010 (3)
- December 2009 (8)
- November 2009 (7)
- October 2009 (3)
- September 2009 (3)
- August 2009 (1)
- June 2009 (4)
- May 2009 (5)
- April 2009 (4)
- March 2009 (2)
- February 2009 (1)
- January 2009 (7)
- December 2008 (1)
- October 2008 (2)
- September 2008 (7)
- August 2008 (1)
- July 2008 (1)
- June 2008 (1)
- April 2008 (5)
- February 2008 (4)
- January 2008 (4)
- December 2007 (3)
- November 2007 (12)
- October 2007 (2)
- September 2007 (4)
- August 2007 (3)
- July 2007 (1)
- June 2007 (3)
- May 2007 (1)
- April 2007 (4)
- March 2007 (3)
- February 2007 (7)
- January 2007 (1)
- December 2006 (2)
- October 2006 (2)
- September 2006 (6)
- July 2006 (4)
- June 2006 (2)
- May 2006 (6)
- April 2006 (3)
- March 2006 (6)

# Category Archives: primes

## More on Fermat witnesses and liars

In my previous post I stated, without proof, the following theorem: Theorem: if is composite and there exists at least one Fermat witness for , then at least half of the numbers relatively prime to are Fermat witnesses. Were you … Continue reading

Posted in computation, number theory, primes
Tagged Carmichael, Fermat, liar, primality, test, witness

## Fermat witnesses and liars (some words on PWW #24)

Let be a positive integer we want to test for primality, and suppose is some other positive integer with . There are then four possibilities: and could share a common factor. In this case we can find the common factor … Continue reading

Posted in computation, number theory, posts without words, primes
Tagged Fermat, liar, primality, test, witness
1 Comment

## The Fermat primality test and the GCD test

In my previous post we proved that if shares a nontrivial common factor with , then , and this in turn proves that is not prime (by Fermat’s Little Theorem). But wait a minute, this is silly: if shares a … Continue reading

## Making the Fermat primality test deterministic

Let’s recall Fermat’s Little Theorem: If is prime and is an integer where , then . Recall that we can turn this directly into a test for primality, called the Fermat primality test, as follows: given some number that we … Continue reading

Posted in computation, number theory, primes
Tagged deterministic, Fermat, primality, test
1 Comment

## Quickly recognizing primes less than 1000: memorizing exceptional composites

In my previous post I wrote about a procedure for testing the primality of any number less than : Test for divisibility by all primes up to , and also . (In practice I test for 2 and 5 first, … Continue reading

## Quickly recognizing primes less than 1000: divisibility tests

I took a little hiatus from writing here since I attended the International Conference on Functional Programming, and since then have been catching up on teaching stuff and writing a bit on my other blog. I gave a talk at … Continue reading