Here's something that I find rather amusing about fractions.
Write each fraction as a decimal:
1/1 =1.000000... Repeats in 1
1/2 =0.500000... Repeats in 1
1/3 =0.333333... Repeats in 1
1/4 =0.2500000... Repeats in 1
1/5 =0.2000000... Repeats in 1
1/6 =0.1666666... Repeats in 1
1/7 =0.142857142857142857... Repeats in 6
1/8 =0.1250000000... Repeats in 1
1/9 =0.111111111... Repeats in 1
1/10=0.1000000... Repeats in 1
1/11=0.09090909090909.... Repeats in 2
1/12=0.0833333333333... Repeats in 1
1/13=0.076923076923076923... Repeats in 6
1/14=0.0714285714285714285... Repeats in 6
1/15=0.0666666666666... Repeats in 1
1/16=0.0625000000000... Repeats in 1
1/17=0.05882352941176470588235294117647... Repeats in 16
1/18=0.055555555555... Repeats in 1
1/19=0.052631578947368421052631578947368421... Repeats in 18
The point is that if p is a prime then
1/p expressed as a decimal will have a repeating pattern of
length N where N divides p-1.
[More generally if a number n=p(1)xp(2)...xp(k) then 1/n written as a decimal will have a repeating
pattern of length N where N divides (p(1)-1)(p(2)-1)...(p(k)-1). See for example 1/14.]
The interesting question is: which primes p do you get expansion where the repeating pattern of length N is equal to (p-1), the maximum length of a repeating pattern?
In the examples above this happens for 7, 17 and our prime 19. One reason I like the 19th step. Some of the primes after this for which you get this maximal repeating pattern are: 23, 29, 47, 59, 61, 97, 109.
Gauss made the conjecture that there are infinitely many primes for which this is true.
This is still unknown. We know that if the extended Riemann Hypothesis is true, then there are infinitely many such primes.
It relates to something called Artin's conjecture. Artin was a mathematician who did his mathematics in the 20s and 30s. I rather liked the following quote I found from Artin:
"We all believe that mathematics is an art. The author of a book, the lecturer in a classroom tries to convey the structural beauty of mathematics to his readers, to his listeners. In this attempt, he must always fail. Mathematics is logical to be sure, each conclusion is drawn from previously derived statements. Yet the whole of it, the real piece of art, is not linear; worse than that, its perception should be instantaneous. We have all experienced on some rare occasions the feeling of elation in realising that we have enabled our listeners to see at a moment's glance the whole architecture and all its ramifications."
I was quite struck by this statement that mathematics was non-linear. This sense of suddenly seeing what was going on is quite true. One might proceed through a linear, logical argument. But then there is this moment when you suddenly see what's going on. Like someone switching on a light and revealing the architecture of the room. Does music have that moment? A non-linear experience amongst the very linear nature of listening to a piece of music?