Hardy and Wright, Chapter 7

Another week, another chapter (last week’s chapter). This week was “General Properties of Congruences”, a reasonably fun chapter. On a side note, I’d like to say how pleased I am that our little group is still going. And note that having a scheduled blog post every week has been a fun habit. It also makes the tags for these post dominate my tag-cloud. I guess that means I should try to spend more time writing about other things as well.

Chris mentioned, before our meeting, how he thought some of the statements were imprecise. I disagreed, but we both thought that some of the wording could be improved. I think that’s to be expected from a book first written 70 years ago. Perhaps I should try to dig up some specific examples.

Chris and Eric and I all agreed that the beginning of the chapter, the first few sections on results about polynomials mod $p$, brought us back to our first year, graduate-level algebra class.

We discussed how the numbers $A_l$, defined to be “the sum of the products of $l$ different members of the set $1,2,\ldots,p-1$” was just the coefficient of $x^l$ in the polynomial $(x-1)(x-2)\cdots (x-(p-1))$. Chris pointed out that really it was the absolute value of that coefficient. If I remember correctly, this was some of the wording Chris was unhappy with.

Most of the rest of our time was spend tracing through the discussion in the text from sections 7.7 and 7.8 on “The residue of $\{\frac{1}{2}(p-1)\}!$” and “A theorem of Wolstenholme”. We all seemed to think that these were fun sections.

As none of the three of us spend much time actually working with integers, we had a bit to discuss with the “associate” of a number mod $p$, or mod $p^2$. We all realized that the associate was just the multiplicative inverse, but had to stop for a second to think about the difference between the inverse mod $p$ and mod $p^2$. We realized, before too long, that if the inverse of $i$, mod $p$, is $n$, then the inverse of $i$, mod $p^2$, has the form $n+p\cdot k$.

We thought about the relationship between integers mod $p$ and mod $p^2$. Thinking mod $p^2$, we write down a $p$ by $p$ array, the first row being $0,1,2,\ldots,p-1$, the next row being $p,p+1,\ldots,2p-1$, and on until the final row, $p(p-1),\ldots p^2-1$. Thinking about multiplicative inverses again, if the inverse of $i$ is $n$ mod $p$ (as in the previous paragraph), then the inverse of $i$ mod $p^2$ appears in the same column as $n$ in this array we have constructed. We were trying to interpret Wolstenholme’s theorem (the sum $1+\frac{1}{2}+\cdots+\frac{1}{p-1}$ is equivalent to 0 mod $p^2$, where $1/i$ is the multiplicative inverse of $i$ mod $p^2$) as summing across rows in this array. That’s not quite right, because the inverses don’t all lie in a row, they are scattered around. If I’m thinking about things correctly, though, these inverses, $1,\frac{1}{2},\ldots,\frac{1}{p-1}$ (mod $p^2$) should occur one in each column of the array we have made (I guess, besides the first column, which is the column of multiplies of $p$).

We didn’t make it to the theorem of von Staudt, concerning Bernoulli numbers taken mod 1. Perhaps we’ll return to it sometime later.