**The Twelve Days of Christmas and Tetrahedral Numbers**

- The Twelve Days of Christmas and Tetrahedral Numbers
- Computing tetrahedral numbers
- Triangular number formula (Challenge #8)
- Binomial coefficients
- Challenge #8 solution
- Binomial coefficients and Pascal’s triangle
- Tetrahedral numbers, exposed!

**Fibonacci numbers and the golden ratio**

- Golden ratio properties (Challenge #10)
- Challenge #10 Solution
- Golden powers
- Explicit Fibonacci numbers

**Perfect numbers**

- Perfect numbers, part I
- Perfect numbers, part II
- Perfect numbers, interlude (Challenge #11)
- Perfect numbers, part III

**Recounting the Rationals**

- Recounting the Rationals, part I
- Recounting the Rationals, part II (fractions grow on trees!)
- Recounting the Rationals, part III
- Recounting the Rationals, part IV
- Recounting the Rationals, part IVb: the Euclidean Algorithm
- Challenge #12: sums of powers of two
- Challenge #12 solution, part II
- More hyperbinary fun
- Hyperbinary conjecture seeking proof for a good time, long walks on the beach
- The hyperbinary sequence and the Calkin-Wilf tree

**Rational and irrational numbers**

- Rational and irrational numbers
- Rational numbers and decimal expansions
- Decimal expansion zoo
- More on decimal expansions
- More on repetend lengths

**Predicting Pi**

**Irrationality of pi**

- Irrationality of pi
- Irrationality of pi: the unpossible function
- Irrationality of pi: derivatives of f
- Irrationality of pi: curiouser and curiouser
- Irrationality of pi: the impossible integral
- Irrationality of pi: the integral that wasn’t

**Decadic numbers**

- A curiosity
- An invitation to a funny number system
- What does “close to” mean?
- The decadic metric
- Infinite decadic numbers
- More fun with infinite decadic numbers
- A self-square number
- u-tube
- Computing with decadic numbers

**A combinatorial proof**

- Differences of powers of consecutive integers
- Differences of powers of consecutive integers, part II
- Combinatorial proofs
- Making our equation count
- How to explain the principle of inclusion-exclusion?
- PIE Day
- Post without words #16
- Post without words #17
- Post without words #18
- Post without words #19
- A probability puzzle
- Probabilistic PIE
- Have a piece of PIE
- Formal PIE
- PIE: proof by algebra
- PIE: proof by counting
**More to come…**

**What I Do**

- What I Do: Part 0
- What I Do, Part 1: Programming languages
**More to come…**

**MaBloWriMo 2015: The Lucas-Lehmer Test and Group Theory**

- The Lucas-Lehmer test
- Mersenne composites
- Mersenne composites in binary
- not all prime-index Mersenne numbers are prime
- The Lucas-Lehmer Test
- The Proof Begins
- s via omega
- definition of s and mod
- omega and its ilk
- Groups
- Examples of Groups
- Groups and Order
- Elements of finite groups have an order
- Element orders are no greater than group size
- One more fact about element orders
- Recap and outline
- X marks the spot
- X is not a group
- groups from monoids
- the group X star
- the order of omega, part I
- the order of omega, part II
- contradiction!
- Bezout’s identity
- Subgroups
- Left cosets
- From subgroups to equivalence relations
- Equivalence relations are partitions
- Equivalence classes are cosets
- Cyclic subgroups

**The chocolate bar game**

- Visualizing nim-like games
- The chocolate bar game
- The chocolate bar game: losing positions in binary
- The chocolate bar game: losing positions characterized
- The chocolate bar game: losing positions proved
- The chocolate bar game: variants

**Fibonacci numbers and golden numbers**

- Testing Fibonacci numbers
- Testing Fibonacci numbers: the proofs
- Fibonacci numbers are golden
- Golden numbers are Fibonacci

**Apollonian gaskets and Descartes’ theorem**

- Post without words #6
- Apollonian gaskets
- Apollonian gaskets and Descartes’ Theorem
- Apollonian gaskets and Descartes’ Theorem II

**Roots of unity and the Möbius function**

- Post without words #10
- A few words about PWW #10
- Post without words #11
- Totient sums
- Complexifying our dots
- Complex multiplication and roots of unity
- Complex multiplication: proof
- Primitive roots of unity
- Sums and symmetry
- Sums of primitive roots
- Computing sums of primitive roots
- The Möbius function
- The Möbius function proof, part 1
- The Möbius function proof, part 2 (the subset parity lemma)
- Dirichlet convolution
- Dirichlet convolution and the Möbius function
- Möbius inversion
- More fun with Dirichlet convolution
- Dirichlet generating functions
- The Riemann zeta function
- The Basel problem
- The MacLaurin series for sin(x)
- The Riemann zeta function and prime numbers

**The greedy coins game**

- The greedy coins game
- Ties in the greedy coins game
- Greedy coins game update
- Another greedy coins game update
- Computing optimal play for the greedy coins game
- Computing optimal play for the greedy coins game, part 2
- Computing optimal play for the greedy coins game, part 3
- Computing optimal play for the greedy coins game, part 4

**The silver ratio**

- The curious powers of 1 + sqrt 2
- The curious powers of 1 + sqrt 2: conjecture
- The curious powers of 1 + sqrt 2: a clever solution
- The curious powers of 1 + sqrt 2: recurrences
- To be continued…?

**Primality testing**

- New baby, and primality testing
- Four formats for Fermat
- Four formats for Fermat: correction!
- Fermat’s Little Theorem: proof by modular arithmetic
- Euler’s Theorem: proof by modular arithmetic
- Fermat’s Little Theorem: proof by necklaces
- Fermat’s Little Theorem: proof by group theory
- A tale of three machines
- Fast and slow machines
- The Fermat primality test
- Modular exponentiation
- Modular exponentiation by repeated squaring
- Post without words #22
- Efficiency of repeated squaring
- Efficiency of repeated squaring: proof
- Efficiency of repeated squaring: another proof
- Primality testing: recap
- Quickly recognizing primes less than 100
- Quickly recognizing primes less than 1000: divisibility tests
- Quickly recognizing primes less than 1000: memorizing exceptional composites
- Making the Fermat primality test deterministic
- The Fermat primality test and the GCD test
- Post without words #24
- Fermat witnesses and liars (some words on PWW #24)
- More on Fermat witnesses and liars
- To be continued…

**Orthogons**

- Post without words #21
- Orthogonal polygons
- Properties of orthogons I
- Properties of orthogons II
- Orthogons and orthobraces
- Listing orthobraces
- Haskell code to naively list orthobraces
- Efficiently listing orthobraces
- Why drawing orthogons is hard
- Drawing orthogons with an SMT solver
- SMT solutions

**The chromatic number of the plane**

- The chromatic number of the plane, part 1
- The chromatic number of the plane, part 2: lower bounds
- The chromatic number of the plane, part 3: a new lower bound
- The chromatic number of the plane, part 4: an upper bound
- Post without words #22
- Some words on PWW #22
- Chromatic number of the plane roundup

**Prefix and repetend length**

- Finding prefix and repetend length
- Finding the prefix length of a decimal expansion
- Finding the repetend length of a decimal expansion
- Post without words #25
- A few words about PWW #25
- More words about PWW #25: The Chinese Remainder Theorem
- Chinese Remainder Theorem proof
- Post without words #26
- Computing the Euler totient function, part 1
- Computing the Euler totient function, part 2: seeing phi is multiplicative
- Computing the Euler totient function, part 3: proving phi is multiplicative
- Computing the Euler totient function, part 4: totient of prime powers

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