# Why was 2 not included in the 'incomposite' numbers in Threshold?

One day I heard two Magi briefly mention the numbers one, three, five, seven, eleven. They are another progression, perhaps.

Boaz tells her:

There is only one thing those numbers have in common. [...] They are all incomposite numbers, except the One, of course, which exists outside and beyond the others. [...] Incomposite numbers are those which cannot be factored - they cannot be divided except by themselves or by the One. They are thus indivisible.

I believe the "incomposite numbers" are what we would call prime numbers (i.e. numbers that aren't in the set of composite numbers). However, 2 is the first prime number, not 3, which is the second.

Most of the people killed by Threshold aren't important characters in the story - they're just there to make up the numbers. Not all of the events themselves are important, either - some simply increase the tension and move us towards the larger and larger numbers that are coming. It didn't seem like killing one extra slave (to start with two) or having another pair of deaths (to continue with two after "the One") would make a significant difference to the story.

Douglass appears to be quite careful with other mathematical aspects of this story, so it doesn't seem like missing 2 would be a simple error. Is there some explanation as to why the sequence didn't either start with two or continue with two after the special "One"?

(I'm fine with an in-universe explanation, if there's one I missed, or an out-of-universe one if Douglass has commented on this somewhere).

• Interestingly, the description is of primes, but incomposite should include 1 naturally. Jul 1, 2012 at 13:58
• At least in the bits you've quoted, it doesn't say that the numbers are all of the "incomposite" numbers (up to a specific point), just that they all are "incomposite." So it's not, strictly speaking, wrong (except for the part where there are lots of other things those numbers have in common: oeis.org/…). Jul 31, 2012 at 6:48

Given that the quote excludes "1" as a special case, a skim through Wikipedia's List of Prime Numbers reveals the "regular prime" classification:

In number theory, a regular prime is a prime number p > 2 that does not divide the class number of the p-th cyclotomic field.

The first few regular primes are:

``````3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, ...
``````

And as a quick note, it doesn't specially exclude 2. There are also odd primes that are excluded from this definition: 37, 59, 67, 101, 103, 131, 149, ...

• The book quote describes quite precisely the normal set of primes (into which 2 fits) and says nothing about cyclotomic fields. Jul 1, 2012 at 13:56
• @Kevin But would the average reader understand cyclotomic fields? It could be something the writer wanted to do, but the editors didn't want to be too complicated. Just a guess, though. Jul 1, 2012 at 16:58
• "And as a quick note, it doesn't specially exclude 2." Yes it does.
– user1030
Jul 31, 2012 at 10:54
• @JoeWreschnig In general, when dealing with complex equations, limitations are imposed at some point to keep them making sense. To borrow an example from another field, the Lorentz factor imposes the limit that we can't go faster than the speed of light - it looks arbitrary until you take a closer look at the equation. While I don't understand cyclotomic fields myself, I figured that particular limit in the definition is required for some similar reason. Jul 31, 2012 at 12:11
• @Izkata: "It excludes 2 because when you plug in 2 the equation doesn't make sense" isn't the same as "it doesn't specially exclude 2." When the definition says "is a prime number p > 2", that's specially excluding 2.
– user1030
Jul 31, 2012 at 14:11

One possible explanation is that "incomposite" is a special subset of primes that excludes 2 - similar to the difference between the natural numbers and whole numbers (the most common definition of whole numbers includes 0, while the most common definition for natural numbers does not).

There are two possible reasons for excluding 2: a) It is the only even prime; b) It is special in that 1 and itself are the positive integers that are less than or equal to it - there are no other possible factors that could make it non-prime.

Whatever the actual reason, there would be some actual purpose for defining the "incomposite" numbers in that way. The reason likely involves some specialized alien math and would almost certainly be outside the scope of a network SF TV series.

• This is a book, not a TV series :) Sep 20, 2012 at 21:53