## Hardy and Wright, Chapter 5

As per schedule, we talked about chapter 5 from Hardy and Wright’s Number Theory book this week (chapter 4 last week). “We” was the smallest it’s been, and so “talked about” was the most brief of our talks to date (besides, perhaps, the organizational meeting).

Chapter 5, “Congruences and Residues”, was, in my mind, a somewhat odd chapter. The first handful of definitions and results, about congruence mod m, I feel like I’ve (mostly) seen several times, so I take it as pretty basic (which isn’t necessarily to say easy to read). I also like thinking about greatest common divisors, which were introduced in this chapter, in the context of category theory, but still am not sure where to go with it. I had hoped, fleetingly, that some lemma in this section might be stated in terms of some lemma about limits or so, but didn’t notice one. Anyway, after this introduction, some crazy sums are mentioned (Gauss’, Ramanujan’s, and Kloosterman’s), but I don’t see the motivation, and they aren’t used. Looking at the index, it looks like only Ramanujan’s will show up later in the text. After this section on sums, the chapter ends with a proof that the 17-gon is constructable. All in all, it seems like a hodge-podge chapter, both in topic and difficulty.

Chris and I agree that using $\equiv$ for “is congruent to” and “is logically equivalent to” is pretty obnoxious. Especially when used all in the same line. For example:

$t+ym'\equiv t+zm'\pmod{p}\equiv m|m'(y-z)\equiv d|(y-z)$

Chris brought up the result that the congruence $kx\equiv l\pmod{m}$ is soluble iff $d=(k,m)$ divides $l$, in which case there are $d$ solutions (this is Theorem 57 in the book). He brought it up because he wasn’t entirely aware of the generalization past $d=1$. I’m not sure I was either, but it is reasonable.

We did agree that using the notation $\overline{x}$ for the solution $x'$ to the equation $xx'\equiv 1\pmod{m}$ (when it exists) was a nice choice of notation. Embedding the integers mod $m$ in the unit circle in the complex plane via $k\mapsto e^{2\pi ik/m}$, the multiplicative inverse of $x$ is the complex conjugate, so the notation lines up.

We finished by talking a little bit about geometric constructions. I liked the long argument about the 17-gon being constructable, by showing that the cosine of an interior angle was constructable, by showing that it was a solution to a system of quadratics (or so). At the beginning of this process, 3 is chosen as a primitive root of 17, and used to define a permutation of $\{0,\ldots,15\}$ by $m\mapsto 3^m\pmod{17}$. I’m not exactly sure why this was done, or why 3 in particular, but it’s fun to trace through the argument from then on. Chris liked the geometric construction, and is tempted to try actually performing it. We reminded ourselves how to bisect angles and draw perpendiculars with straight-edge and compass.