Number Theory

Random Cool Math Thing #4

Visual Quadratic Reciprocity

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Farey Fraction Visual Patterns

The Farey Sequence, \(F_{n}\) is the list of all simplified fractions between \(\dfrac{0}{1}\) and \(\dfrac{1}{1}\) with denominator less than or equal to \(n\). For example,

  \( \large F_{5}=\dfrac{0}{1},\dfrac{1}{5},\dfrac{1}{4},\dfrac{1}{3},\dfrac{2}{5},\dfrac{1}{2},\dfrac{3}{5},\dfrac{2}{3},\dfrac{3}{4},\dfrac{4}{5},\dfrac{1}{1}\)
 
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Pool Table Math with Excel (includes a Video)

This is a classic problem solving activity that I first saw many years ago in Mathematics: A Human Endeavor.   The NCTM has a nice applet to test cases one at a time.  

Since I like playing with Excel and trying to extend what it can do, I wrote a spreadsheet to represent the problem.  The end result is much better than expected — this representation uses a slider to show many cases in a short period of time, and with it I noticed patterns that I never had before (e.g. if you start at (1,0) on a rectangle whose sides have GCD=2, you get a loop that goes through all squares). Read more >>

Primes Less than 100

Sporcle game here.   Name all the primes less than 100 in 3 minutes.