This is a series on the book Gödel, Escher, Bach: An eternal golden braid by Douglas Hofstadter.
Earlier diaries are here
Today, we will examine Aria with diverse variations p. 391 - 405.
After today, the going gets very heavy, and I will be needing some help from computer science types ... so, take the poll!
From the Overview
A dialogue whose form is based on Bach's Goldberg Variations and whose content is related to number-theoretical problems such as the Goldbach conjecture. This hybrid has as its main purpose to show how number theory's subtlety stems from the fact that there are many diverse variations on the theme of searching through an infinite space. Some of them lead to infinite searches, some of them lead to finite searches, while some others hover in between.
On page 393, DH makes a table like this:
6 = 3 + 3
8 = 3 + 5
10 = 3 + 7 = 5 + 5
12 = 5 + 7
14 = 3 + 11 = 7 + 7
16 = 3 + 13 = 5 + 11
Each row is an even number, and the sums of two primes which make that even number. The Goldbach conjecture is one of the oldest unsolved problems in number theory. It is
Every even number can be written as the sum of two primes
Here's an idea that jumped into my mind, looking at this table:
The first number after the + sign never goes down
On p. 394, DH mentions that Schirelman proved that every number is the sum of at most 300,000 primes. That's odd! How does one prove such a thing? Well, according to wikipedia it wasn't 300,000 but 20. Still ... this article is not too much help.
On p 399 there is a picture of order and chaos by Escher. I do not 'get' this picture.
On p. 401 DH gets into what he calls Wondrous Numbers , these seem to have several aliases. The idea:
Start with any positive integer
If it is even, half it
If it is odd, triple it and add one.
So, if you start with, say, 12:
12 even so half =
6 even so half =
3 odd so 3n+1 =
10 even so half =
5 odd so
16 which then goes to
1 which goes back to 4
the conjecture is that every number is wondrous. Every number so far has been found wondrous; the number of steps shows little pattern.
The number under 10000 with the most steps is 27, which takes 111 steps. The number under 100 million with the most steps is 63,728,127 which takes 949 steps.
On p 402, there are analogies between the increasing and decreasing of the wondrous numbers and the lengthening and shortening of the MU strings.
On p 403 starts a loop of self reference from the dialogue to itself. Did the tortoise tell Hofstadter about this?
Then on p 404 is a sort of analogue to Cantor's diagonal proof, only with letters.