With renormalisation, a basic circulant may be written as a matrix $A$ with integral entries. Let us consider powers, such as
More generally, with coefficients $n$ and $n - 1$ we obtain
For example, when the initial coefficient pair is $(3,2)$ the sequence of powers gives coefficient pairs $(j, j - 1)$ that sum to $5^{m}$. That $3$ is odd corresponds to the fact that $5 = 1 \textrm{mod} 4$, since we write $n = 2k + 1$ and the sum is then $4k + 1$. The matrix sequence demonstrates the obvious fact that $5^m = 1 \textrm{mod} 4$ for all $m$, as the first coefficient is always odd.
A number theorist will tell you that an odd prime $p$ equal to $1 \textrm{mod} 4$ satisfies $p = a^2 + b^2$ for unique $a$ and $b$. The integer ratio $a/b$ comes from $(2k)!$. For $p = 5$, $a$ and $b$ are usually found from the Gaussian integer factorization $5 = (2 + i)(2 - i)$, where $a = 2$ and $b = 1$. Each Gaussian factor may be expressed as another basic two crossing braid, as in where the factors also come from the Jordan product. The number $x$ (with $x^2 = 1$) is omitted here, since it always appears in the same places. Note that the square of one such factor multiplies (up to signs) the first coefficient by $3$ and the second by $2$, as in the matrix above. How nice to see products of basic $B_3$ configurations expressing simple arithmetic rules.
14 years ago
Note that the Gaussian factor braid differs only by one crossing flip from the braid in the last post, giving two overcrossings instead of one.
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