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Two Kinds of Odd 
RepunitsA repunit, or repeated unit, is a number consisting entirely of 1s. I don't like the name, but it seems to be standard, so I guess I'm stuck with it. Anyway, I've never been very familiar with repunits, but they've been turning up a lot recently, and I keep forgetting which ones have which factors, so just for reference I thought I'd make a table.
There are some interesting patterns in there, but before I can explain them I'll need to remind you of a couple of things. First of all, when I was talking about fractions in base 2, I observed that repunits of composite length have nice factorizations that are really factorizations of polynomials, and suggested that it would be handy to have a digit that represents 1. Then, when I was talking about divisibility in other bases, I started using the hash mark as such a digit not to be confused with the sharp sign from Sharps and Flats. Knowing all that, we can break the repunits into irreducible polynomial factors without even specifying what the base is!
The irreducible polynomial factors may break down further in particular bases; here's what happens in bases 10 and 2. (I've converted the base 2 results into base 10, hope that's not too confusing. For example, 111 in base 2 is the prime number 7.)
Each repunit contains exactly one new irreducible polynomial I'm not sure why that's true, but it is, and I've numbered the rows accordingly in the table. Now I can explain the main pattern in the original table. Let's call the repunit of length n R_{n}, and the associated irreducible polynomial P_{n}. If n is divisible by some number m, then R_{n} is divisible by R_{m} because there's a nice factorization; and R_{m} in turn is divisible by P_{m}, by definition. So, the numbers that are factors of P_{m} in some base will be factors of R_{n} whenever n is a multiple of m. Actually, that's not quite true it fails for m = 1, because R_{n} is never divisible by P_{1}. And where'd that polynomial P_{1} come from, anyway?! Well, I was planning ahead. Suppose we define one more set of polynomials by the rule B_{n} = R_{n} ื P_{1}; then B_{n}, unlike R_{n}, really is divisible by P_{m} whenever n is divisible by m; and in fact B_{n} is equal to the product of all such P_{m}. That may sound complicated, but the polynomials B_{n} are actually extremely simple. For example,
B_{5} = R_{5} ื P_{1} = (b^{4} + b^{3} + b^{2} + b + 1)(b  1) = b^{5}  1. The same cancellation happens every time, so that B_{n} = b^{n}  1 for all n. Now, the roots of B_{n} are roots of unity, and roots of unity lie on the unit circle in the complex plane, so when we divide B_{n} into the parts P_{m}, we're dividing the circle which is roughly why the parts P_{m} are called cyclotomic polynomials. (The roots of the word are cyclo, as in cyclic, and tomic, as in atomic, indivisible.) Just for fun, here are some pictures of the roots of P_{m}.
See how for example P_{1}, P_{2}, P_{3}, and P_{6} combine to cover the sixth roots of unity? That was fun, but let's get back to the subject of patterns. I think we've said all there is to say about the main pattern, the pattern in how the repunits break down into cyclotomic polynomials, but there are also a few patterns in how the cyclotomic polynomials break down into factors in base 10.
Finally, let me go back and clarify a couple of things. First, the idea of numbers being polynomials over some unspecified base isn't restricted to repunits and cyclotomic polynomials. For example, the number 299 is never prime, because
299 = 2b^{2} + 9b + 9 = (b + 3)(2b + 3), and the number 156 is always an intermediate, because
156 = b^{2} + 5b + 6 = (b + 2)(b + 3) = (b + 2)^{#}. (See also the discussion of small factors and carries in Multiplication.) Second, in spite of the evidence in the table, cyclotomic polynomials don't always consist of evenlyspaced 1s or evenlyspaced alternating 1s and #s against a background of 0s. Some have repeating triplets,
some have other, stranger patterns,
and some even have other digits! According to an article on cyclotomic polynomials, the first one of those occurs at 105 (for the same reason as in Number Maze!), and here it is, with X being a digit that represents 2.
11100##X##0011111100#0#0#0#0#0011111100##X##00111

See AlsoHistory and Other Stuff Multiplication in Base 10 @ May (2006) 