### In The Bleak Midwinter

This is a poem written by Christina Rossetti, set to music first by Gustav Holst in 1906 and then by Harold Darke in 1911. This is by Darke :   © OldTrout

### O Come, O Come, Emmanuel

Today is the first Sunday of Advent.  It is time to light the first candle. #Hope. O come, O come, Emmanuel, And ransom captive Israel, That mourns in lonely exile here, Until the Son of [more…]

### Benford’s Law

Benford’s Law is a theory of leading digits in numerical data sets. Whilst one might think that the frequency of the leading digit would be equally distributed across the nine digits each with approximately 11% [more…]

### The Padovan Sequence and the Plastic Ratio

The golden ratio is well known and is associated with the Fibonacci sequence of numbers.     (phi) is the positive root of the quadratic equation . The exact value is given by Its continued [more…]

### Sophie Germain Primes and Cunningham Chains

A Sophie Germain prime is a prime number such that is also a prime number. That is, ,  where and are both prime.  For example, . The sequence begins Sophie Germain () explored these primes [more…]

### The Infinity of the Primes

Euclid’s Elements, Book IX, Proposition 20. Theorem : Prime numbers are more than any assigned multitude of prime numbers. That is, there are infinitely many primes. Proof : Suppose that there are primes : Let [more…]

### The Number of Derangements

Read as : the limit, as approaches infinity, of subfactorial divided by factorial is equal to divided by which approximates to , where is Euler’s number, The Hat Check Problem, as first described by Montmort [more…]

### On Punctuation!

Apparently, some subset of younger people are upset and intimidated about the use of full stops in text and other messages. I am not sure what they would think about the ellipsis, … , or [more…]

### From a Sum to a Product

In The Basel Problem we saw that Euler answered the question by showing that the sum of the reciprocals of the square numbers is equal to . That is The partial sums slowly converge to [more…]