Newsletter 25: Friday 19 June 2026

For the four MONDAYS 6th, 13th, 20th and 27th in July there will be a
conversation on the FOUNDATION STONE at the centre.
Conversations from 7 pm
The Foundation Stone as laid at the Christmas Conference 1923-24 is
such a fundamental event in the life of Anthroposophy that it is not easy to
comprehend all its ramifications. I would like to consider one aspect:
what it cost RUDOLF STEINER to take on his shoulders, the destiny of
the Anthroposophical Society.

Christopher Bacchus

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Arithmetic Musings #11

Last week we looked at the rarity of perfection in numbers.

The whole numbers can be split into two groups:

a. Prime numbers - the building blocks;

b. Composite Numbers - made up as multiples of Primes, whether Deficient, Perfect, or Abundant

This week we will look at finding Prime numbers by eliminating [sieving out] the composite numbers that are multiples of each Prime.  This was done by an Egyptian savant called Eratosthenes, who lived over 2,200 years ago. He was the Librarian of Alexandria and was famous for measuring the size of the earth without leaving Egypt. 

Firstly. we can consider the multiples of a prime as its family or relatives in the Kingdom of Whole Numbers.  Number 1 is the King of numbers - it rules every whole number.  Neighbours are separated by 1. Now number 2 considered himself to be very important as Prime Minister.  Once he decided to have a Banquet for all his relatives - the multiples of 2.  So invitations were sent out to them in Hundred City.

Here is a map of the city:

In this map, all of 2's relatives are shown in red.  Every second Avenue was full of them.

Well, 3 was not invited, so he decided to have a Banquet just for his relatives - the multiples of 3.  The stepwise pattern they make in orange is very different, and interesting:

As Primes are never invited to another Prime's Banquet, they have to hold their own.  The Abundant numbers have a great time - they are invited to quite a few banquets.  

Here are some more maps showing the patterns that emerge.

When all of this information from Eratosthenes is combined, we can see how differentiated numbers can be:

The first four perfect numbers 6, 28, 496 and 8128 seem to have been known from ancient times - there is no historical record of their discovery.

1^st.         6 = 1 + [2 + 3],
2^nd.        28 = 1 + 2 + [4 + 7] + 14,
3^rd.        496 = 1 + 2 + 4 + 8 + [16 + 31] + 62 + 124 + 248
4^th.        8128 = 1 + 2 + 4 + 8 + 16 + 32 + [64 + 127] + 254 + 508 + 1016 + 2032 + 4064

Notice how the numbers are double the preceding number in the sequence, except for the pair near the middle - the second number is one less than double the first one, and in each case the second of the pair is a prime number.  So here the powers of 2 are very important.

Here is a chart showing the calculations for the first seven perfect numbers.  As you can see they grow very fast, and are fewer and further between.  The seventh one is already greater than a hundred billion.  It would take a lifetime to count as far as this!

 

RB

More in the coming weeks