
Join 714 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)
Tag Archives: order
Order of operations considered harmful
[The title is a halfjoking reference to Edsger Dijkstra’s classic paper, Go To Statement Considered Harmful; see here for more context.] Everyone is probably familiar with the socalled “order of operations”, which is a collection of rules that reflect conventions … Continue reading
Posted in arithmetic, teaching
Tagged harmful, operations, order, parentheses, parsing, perform, trees
9 Comments
Fermat’s Little Theorem: proof by group theory
It’s time for our third and final proof of Fermat’s Little Theorem, this time using some group theory. This proof is probably the shortest—explaining this proof to a professional mathematician would probably take only a single sentence—but requires you to … Continue reading
Posted in group theory, number theory, primes, proof
Tagged combinatorics, group, order, proof, theory
Comments Off on Fermat’s Little Theorem: proof by group theory
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
MaBloWriMo 15: One more fact about element orders
I almost forgot, but there is one more fact about the order of elements in a group that we will need. Suppose we have some and we happen to know that is the identity. What can we say about the … Continue reading
Posted in algebra, group theory, number theory, proof
Tagged elements, finite, groups, MaBloWriMo, order
1 Comment
MaBloWriMo 14: Element orders are no greater than group size
Today we will give an answer to the question: What is the relationship between the order of a group and the orders of its elements? Yesterday, I claimed we would prove that for any element of a group , it … Continue reading
Posted in algebra, group theory, proof
Tagged elements, finite, groups, MaBloWriMo, order
2 Comments
MaBloWriMo 13: Elements of finite groups have an order
Recall from yesterday that if is a group and is some element of the group, the order of is defined as the smallest number of copies of which combine to yield the identity element. I forgot to mention it yesterday, … Continue reading
Posted in algebra, group theory, proof
Tagged elements, finite, groups, MaBloWriMo, order
Comments Off on MaBloWriMo 13: Elements of finite groups have an order
MaBloWriMo 12: Groups and Order
Continuing our discussion of groups (see here and here), today I want to discuss the concept of order, which is defined both for groups themselves and for the elements of a group. The order of a group simply means the … Continue reading
A computerchecked proof of the odd order theorem
Big news: a proof of the FeitThompson Theorem (also known as the “odd order theorem”) has been completely formalized and verified by a computer, using the Coq proof assistant! Wait, what? Huh? you’re probably thinking. Well, let me unpack that … Continue reading