Discrete Logarithms

In the fall of 2017, I created what I propose to consider as being both art and mathematics. Call the art conceptual; the mathematics, expository; here it is, as a booklet of 88 pages, size A5, in pdf format.

More precisely, the work to be considered as both art and mathematics is the middle of the three chapters that make up the booklet. The first chapter is an essay on art, ultimately considering some examples that inspire my own. The last chapter establishes the principle whereby the lists of numbers in Chapter 2 are created.

Each of the lists is of the same ordered pairs of integers. There are a thousand pair, less five, in five columns on each page. For each pair (nk) on the lists,

1 < n < 997,

0 < k < 996, and

7kn (mod 997).

The last expression is read as, “7 to the power k is congruent to n modulo 997”; it means 7kn is a multiple of 997. One can say then that k is the (discrete) logarithm, to the base 7, of n modulo 997; and one can write

k ≡ log7 n (mod 996).

The first list of the pairs (nk) is according to n; the second, k. One can use the lists in tandem to turn multiplication problems into addition problems, because of one of the laws of exponents, also expressed as a law of logarithms:

bkb = bk + ,

logb (nm) = logb n + logb m.

The good old-fashioned slide rule exploits these laws. A basic slide rule is like two rulers, each measuring 10 inches; but the number printed k inches from the beginning is 10k/10, so that k/10 is the base-10 logarithm of this number. One multiplies the printed numbers by adding their distances from the beginning, which is marked 1, not 0.

Before the advent of the pocket calculator, the slide rule had practical value. It may still be useful for learning about logarithms, and logarithmic scales can be useful for graphing such things as Covid-19 cases (as on my friend’s handy site).

To show my students, I actually used discrete logarithms to make slide rules (such as the circular slide rule depicted on the cover of my booklet). I don’t however think my tables of discrete logarithms have any practical value. Thus they may be Chindōgu.

My tables can exist because

  1. 997 is a prime number;

  2. prime numbers have “primitive roots”;

  3. 7 is a primitive root of 997.

I spent much of the fall semester of 2017, proving the theorem that prime numbers have primitive roots. This was in my department’s first-year number-theory course. Thus Chapter 3 of my booklet concentrates a lot of work for the beginner into a few pages. The chapter also looks at the historical origins of some of our terminology, such as “modulo” (which is not actually in any of the print dictionaries in my personal library).

For students in my 2017 course, I created a summary, “Sayılar Kuramına Giriş Özeti” (5 pages, size A4, two columns, landscape orientation, in our language of instruction, which is Turkish).

Chapter 3 of the booklet begins by reviewing Euclid’s proof of the commutativity of multiplication of counting numbers; I had taken this up also, earlier in 2017, in a blog post, “The geometry of numbers in Euclid.” I had once considered trying to teach number theory as Euclid did; but I decided this would be too strange.

In 2013, I had learned of a conceptual work by Claude Closky, “The first thousand numbers classified in alphabetical order.” It is what it is called. I translated it into Turkish as “Alfabe sırasına göre sınıflanmış ilk bin sayı” (8 pages, size A5), and I created for this document a page of my departmental website. Now I am delighted to see that my page is linked to by what appears to be Closky’s own page.

Also linked to from that page is “The First M Numbers Classified in Alphabetical Order,” by Nick Montfort (2013); the work is a program to list the first thousand Roman numerals in alphabetical order.

Not knowing about that, I made such a list myself, later that year, up to MMMCMXCIX; the result, “The Roman numerals in alphabetical order,” is 165 pages of size A5. Like the discrete logarithms, the list of Roman numerals may also be Chindōgu, because in principle one could use it to interpret such numerals, but nobody ever would; at least nobody should; but then my friend once said she had given her secretary a chart as a reminder that 1/2 = 0.5 and 1/4 = 0.25.

I am pleased to have at hand all of the books I used in preparing my booklet. Some I have only as electronic files; I photographed the others for this post. Then I realized that I had left out the book on Duchamp. I noticed that its cover was complemented by that of Rabih Alameddine, An Unnecessary Woman (New York: Grove Press, 2013). The woman of the title is useless for making translations that nobody ever reads.

Apparently I read her story in 2015, since the receipt tucked inside the novel is dated to April of that year. Another piece of paper tucked inside is a cinema ticket, marking the page of a nice paragraph:

In a silly essay on Crime and Punishment, a critic suggests that Raskolnikov is the epitome of the Russian soul, that to understand him is to understand Russia. Tfeh! Not that the proposition isn’t true; it may or may not be. I’ve yet to meet Russia’s soul. What the reviewer is doing is distancing himself from the idea that he too is capable of killing a pawnbroker. We’re supposed to infer that only someone with a Russian soul would.

Some art is good, precisely for not distancing you, but giving you both the idea that you can do it for yourself, and the impetus to do it.

2 Trackbacks

  1. By Re-enactment « Polytropy on August 15, 2020 at 6:19 am

    […] This and other examples show that, in its article on Aesthetics by Barry Hartley Slater, the Internet Encyclopedia of Philosophy is misleading to suggest that Collingwood “took art to be a matter of self-expression.” There was no call to add the restriction to the self. (I pursue this a bit further in § 1.3, “Individualism,” of the booklet Discrete Logarithms: Mathematics and Art, introduced in a post of the same name, “Discrete Logarithms.”) […]

  2. By What Mathematics Is « Polytropy on September 15, 2020 at 9:13 am

    […] procedure in the pdf booklet that is linked to from the post about art and mathematics called “Discrete Logarithms.” In the middle of his book, Collingwood offers the […]

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