
Join 713 other subscribers
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: modular arithmetic
Post without words #26
Posted in modular arithmetic, number theory, posts without words
Tagged Euler, grid, totient
2 Comments
Chinese Remainder Theorem proof
In my previous post I stated the Chinese Remainder Theorem, which says that if and are relatively prime, then the function is a bijection between the set and the set of pairs (remember that the notation means the set ). … Continue reading
More words about PWW #25: The Chinese Remainder Theorem
In a previous post I made images like this: And then in the next post I explained how I made the images: starting in the upper left corner of a grid, put consecutive numbers along a diagonal line, wrapping around … Continue reading
Posted in modular arithmetic, number theory, posts without words
Tagged Chinese, grid, remainder, theorem, torus
3 Comments
A few words about PWW #25
In my previous post I made images like this: What’s going on? Well, first, it’s easy to notice that each grid starts with in the upperleft square; is one square down and to the right of , then is one … Continue reading
Posted in modular arithmetic, number theory, posts without words
Tagged Chinese, grid, remainder, theorem, torus
4 Comments
Finding the repetend length of a decimal expansion
We’re still trying to find the prefix length and repetend length of the decimal expansion of a fraction , that is, the length of the part before it starts repeating, and the length of the repeating part. In my previous … Continue reading
Posted in computation, group theory, modular arithmetic, number theory, pattern
Tagged decimal, expansion, group theory, rational, repeating, repetend, totient
Comments Off on Finding the repetend length of a decimal expansion
MaBloWriMo 24: Bezout’s identity
A few days ago we made use of Bézout’s Identity, which states that if and have a greatest common divisor , then there exist integers and such that . For completeness, let’s prove it. Consider the set of all linear … Continue reading
Posted in algebra, arithmetic, modular arithmetic, number theory
Tagged Bezout, combination, divisor, gcd, identity, linear, MaBloWriMo, proof
2 Comments
MaBloWriMo 23: contradiction!
So, where are we? We assumed that is divisible by , but is not prime. We picked a divisor of and used it to define a group , and yesterday we showed that has order in . Today we’ll use … Continue reading
Posted in algebra, group theory, modular arithmetic, number theory, proof
Tagged contradiction, groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, order, prime, proof, test, X
5 Comments
MaBloWriMo 22: the order of omega, part II
Yesterday, from the assumption that is divisible by , we deduced the equations and which hold in the group . So what do these tell us about the order of ? Well, first of all, the second equation tells us … Continue reading
Posted in algebra, group theory, modular arithmetic, number theory, proof
Tagged groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, order, prime, proof, test, X
1 Comment
MaBloWriMo 21: the order of omega, part I
Now we’re going to figure out the order of in the group . Remember that we started by assuming that passed the LucasLehmer test, that is, that is divisible by . Remember that we also showed for all . In … Continue reading
Posted in algebra, group theory, modular arithmetic, number theory, proof
Tagged groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, order, prime, proof, test, X
2 Comments