How’s 2 million digits, then?
Okay, the 2 million digits of e helped a lot. 
I’ve now got an e Spirals page, and also a page showing a couple of ‘extreme’ random spirals.
Looking at the histograms from the random spirals, it appears that it might be the case that low-linearity spirals have more (relatively) big gaps between two neighboring dots along the spiral. The low- and mid-linearity random spirals have much larger right-hand columns (63-plus black dots between two white dots) than the high-linearity spiral does.
Of course, all these histograms are “normalized” to the highest bar going top-to-bottom, regardless of its absolute value, so it may be an artifact of the lower-linearity histograms having smaller peak values, instead.
Actually, that’s not it. The histograms on the e page for digits 1 (lower linearity) and 3 (higher linearity) are backwards from what I would have expected were that guess correct. More testing might reveal something else.
Oh, I only spent an hour-and-a-half on these two new sets of images and pages (note the similarities between the pi page and the e page - cut-and-paste is wonderful), and none of that time included fixing my linearity calculators (but it did include taking out the garbage and cleaning up the kitchen), so these values are still low, but they’re also still comparable to the older two pages.
Fixing that bug is going to have to wait for a larger chunk of available time, perhaps this-coming weekend. And when that’s fixed, I’ll go repair the two older pages with the correct numbers, as well.
Dave, I must applaud you efforts! Way to go. This is some pretty cool stuff, even though there’s not a great deal to be learned. It does seem to add weight to the speculation that the digits of pi and e are randomly distributed, though.
it is possible that the digits of pi and e are not random, but that we just use the wrong numerical system to express them.
what if you used say a base 12 or base 70 system? wouldn’t the “decimals” (or rather “duodecimals” and “septantesimals”) look quite different?
Afraid not. If pi and e and transcendental in base 10, then they are transcendental in every base. It is an intrinsic property, not a function of the way they happen to be written down.
Transcendental isn’t the property you’re looking for. There are some trancendental numbers whose representation in a particular base is by no means random.
Normality is the property of interest. pi is believed to be normal in every base, although no one’s proven it yet, IIRC. I don’t know anything about e.
I have an interesting take on coloring the Ulam Spiral, using DaveW’s method of multiple shades of grey:
Brightest for Primes, next brightest for product of two primes, second next brightest for product of three primes, etc.
One of the things I’ve contemplated about numbers is that you don’t want a “softvark” definition (A “softvark” is anything that isn’t an 'aardvark ;)) – you should not define a category by what it is not, but by what it is. No negative words, in other words, should enter into your definition.
Given that, a prime number is defined as a natural number which two and only two discrete factors (itself and one).
However, this produces an interesting bit of categorization:
The only natural number with only one factor is one.
The only numbers with precisely three discrete factors are the squares of primes: their factors are one, the prime, and the square itself.
Composites produced by multiplying two discrete primes have exactly four factors: one, the first prime, the second prime, and their product. But the cubes of primes also have exactly four factors: one, the prime, its square, and its cube.
The fourth power of a prime has exactly five factors; the product of a four-factor composite and a prime has exactly six (as does the fifth power of a prime).
In this system, instead of being the peculiar (and therefore interesting) leftovers of number manipulation, primes become the basis of the system.
The first twenty natural numbers, sorted by this system:
Number No.Factors Factors
1 1 1
2 2 1, 2
3 2 1, 3
4 3 1, 2, 4
5 2 1, 5
6 4 1, 2, 3, 6
7 2 1, 7
8 4 1, 2, 4, 8
9 3 1, 3, 9
10 4 1, 2, 5, 10
11 2 1, 11
12 6 1, 2, 3, 4, 6, 12
13 2 1, 13
14 4 1, 2, 7, 14
15 4 1, 3, 5, 15
16 5 1, 2, 4, 8, 16
17 2 1, 17
18 6 1, 2, 3, 6, 9, 18
19 2 1, 19
20 6 1, 2, 4, 5, 10, 20