Christianity
The Gifts of Christmas
If you sent all the gifts in the Twelve Days of Christmas, then how many would you send? The gifts for each day are the triangular numbers : We have done these before : In [more…]
Counting
Hobbies
The Möbius Function
The Möbius function is denoted by , where is the Greek letter mu. Depending on the factorisation of , the function takes the values . The last says that if is even, then and if [more…]
Hobbies
The Orbit-Stabiliser Theorem
Some notation and definitions. is a finite group. is the order (size) of a group. is a finite set. is the order (size) of a set. Group action : For an action of the group [more…]
Hobbies
Proof by Necklaces
A proof of Fermat’s Little Theorem using necklaces. , where is prime and is any integer with gcd. Multiply both sides of the congruence by : Which gives : Subtract from both sides : [more…]
Hobbies
The Confusing Bit
When you do mathematics you spend much of your time being confused. We can go further with our counting and count the number of necklaces and bracelets with a particular colouring. When we counted the [more…]
Hobbies
The Dihedral Group of a Bracelet
A Dihedral group, , is the group of rotations and reflections (the symmetries) of a regular -gon. The order (size) of the group is . We considered the rotations in the Cyclic group of [more…]
Hobbies
The Cyclic Group of a Necklace
A Cyclic group is a group that can be generated by a single element . The generator is denoted by and satisfies : , where is the identity element. A Cyclic group, , is the [more…]
Hobbies
Euler’s Totient Function
Euler’s totient function counts the number of positive integers up to that are relatively prime to , where is considered to be relatively prime to all . These numbers are called the totatives of . [more…]