Hobbies
Distinct and Odd Partitions
Euler’s Partition Theorem: The number of distinct partitions of an integer, , is equal to the number of odd partitions of that integer : , for all . Euler gave us a proof which is [more…]
Leonhard Euler
Computing
Computing Partitions
Euler gave us the following recursion formula for via his Pentagonal Number Theorem : Note the generalised pentagonal numbers in the above. Define if and let . This is the formula that [more…]
Hobbies
Unrestricted Partitions
Some may have noticed that I have been giving a representation of the countdown clock. The first line is the fundamental theorem of arithmetic. That is, every integer can be represented by a unique factorisation [more…]
Hobbies
Euler’s Pentagonal Number Theorem
Euler, : In considering the partitions of numbers, I examined, a long time ago, the expression in which the product is assumed to be infinite. In order to see what kind of series will result, [more…]
Hobbies
Generalised Pentagonal Numbers
We considered the pentagonal numbers which begin : That is : for Now consider the following sequence : which begins at and has a common difference of between the terms. For any arithmetic sequence, we [more…]
Hobbies
On Pentagonal Numbers
Hypsicles of Alexandria (c.190 – c.120 B.C.) gave the following definition of polygonal numbers : If there are as many numbers as we please beginning from 1 and increasing by the same common difference, then, [more…]
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The Basel Problem
One sometimes gets very interesting results when one turns divergent sums upside down. The Basel problem was asked by Pietro Mengoli in . He asked what is the exact summation of the reciprocals of the [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
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…]