
Join 712 other followers
Meta
Categories
 algebra (47)
 arithmetic (86)
 books (35)
 calculus (7)
 challenges (59)
 combinatorics (31)
 complex numbers (6)
 computation (83)
 convergence (9)
 counting (38)
 famous numbers (49)
 fibonacci (18)
 fractals (13)
 games (34)
 geometry (73)
 golden ratio (8)
 group theory (28)
 humor (8)
 induction (8)
 infinity (19)
 iteration (24)
 links (77)
 logic (12)
 meta (43)
 modular arithmetic (30)
 number theory (108)
 open problems (11)
 paradox (1)
 pascal's triangle (8)
 pattern (106)
 people (23)
 pictures (74)
 posts without words (44)
 primes (57)
 probability (9)
 programming (20)
 proof (93)
 puzzles (18)
 recursion (16)
 review (25)
 sequences (28)
 solutions (31)
 teaching (16)
 trig (3)
 Uncategorized (6)
 video (19)
Archives
 August 2021 (2)
 June 2021 (3)
 May 2021 (1)
 March 2020 (4)
 February 2020 (1)
 January 2020 (7)
 December 2019 (4)
 November 2019 (2)
 October 2019 (5)
 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: arithmetic
Computing the Euler totient function, part 2: seeing phi is multiplicative
In my last post, I claimed that whenever and are relatively prime. (Recall that counts how many numbers from to share no factors in common with .) Let’s get some intuition for this by looking at some Chinese remainder theorem … Continue reading
Computing the Euler totient function, part 1
Recall that Euler’s totient function counts how many of the integers from to are relatively prime to , that is, share no factors in common with . For example, , since only , , , and share no factors with … Continue reading
The wizard’s rational puzzle (solutions, part 2)
At long last, here is the solution I had in mind for the Wizard’s rational puzzle. Recall that the goal is to figure out the numerator and denominator of a secret rational number, if all we are allowed to do … Continue reading
Posted in arithmetic, challenges, logic, programming, puzzles, solutions
Tagged arithmetic, binary, denominator, Euclidean, gcd, logarithm, numerator, puzzle, rational, search, wizard
Comments Off on The wizard’s rational puzzle (solutions, part 2)
The wizard’s rational puzzle (solutions, part 1)
About two and a half months ago I posted a challenge involving a sadistic math wizard, metal cubes containing rational numbers, and a room full of strange machines. I’ve been remiss in following up with some solutions. (Go read the … Continue reading
Posted in arithmetic, challenges, logic, programming, puzzles, solutions
Tagged arithmetic, denominator, Euclidean, gcd, logarithm, numerator, puzzle, rational, wizard
3 Comments
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
Quickly recognizing primes less than 100
Recently, Mark Dominus wrote about trying to memorize all the prime numbers under . This is a cool idea, but it made me start thinking about alternative: instead of memorizing primes, could we memorize a procedure for determining whether a … Continue reading
Post without words #22
Posted in arithmetic, computation, number theory, posts without words
Tagged logarithmic, repeated, squaring
8 Comments
The wizard’s rational puzzle (mind your p’s and q’s!)
You have been abducted by a sadistic math wizard (don’t you hate it when that happens?). He ushers you into a plain but cozylooking room, with a hardwood floor, a few exoticlooking rugs, and wood paneling on the walls. He … Continue reading
Posted in arithmetic, challenges, logic, programming, puzzles
Tagged arithmetic, denominator, numerator, puzzle, rational, wizard
15 Comments
Iterating squared digit sums in other bases
In a previous post I wrote about iterating the squared digit sum function, which adds up the sum of the squares of the digits of a number; for example, . Denis left a comment asking about other bases—what happens if … Continue reading