It turned out that the class was not yet finished with the number theoretic topics generated by a study of repetends. The author, using Maple V software, had produced a list of prime factors of "Davidson numbers." (See day 4 for the initial discussion of these numbers, which are composed of nothing but a string of 1's.) Below is a table up to size n = 37.































































































































































































































































The author found that subsequent numbers of this form up to at least n = 46 are composite, and these will be included in the table at a later date.
The author also had a bit of skepticism regarding the primality of the largest prime factors generated by the mathematical software Maple V. Consider the case for n = 41. Maple V found that:
Is 201,763,709,900,322,803,748,657,942,361 really prime? Probably. The square root of this number is about 449,000,000,000,000, so determining that 201,763,709,900,322,803,748,657,942,361 is prime involves exausting all primes up to around 449,000,000,000,000. Assuming that roughly 1% of the numbers up to 449,000,000,000,000 are prime (the author believes this to be a conservative estimate), that means around 4,490,000,000,000 divisions to check for primality of 201,763,709,900,322,803,748,657,942,361. The author once wrote a small program in C programming language which determined that his beloved Mac Powerbook 3400 could perform 22 million integer additions per second. Just applying that standard to this problem means that at the very best it would take about 4,490,000,000,000 ÷ 22,000,000 = 204,091 seconds, or about 57 hours, to determine primality, when in fact Maple V did it in less than thirty minutes. (Unsure of exact time.)
By strange coincidence the author received a phone call the next morning from the company which makes Maple, called Waterloo Maple. The company representative was confident his product produced a bonafide prime factorization. Maple uses powerful algorithms written by very competent mathematicians.
Returning to the subject of the socalled "Davidson numbers," we set up a onetoone correspondance from the natural numbers to the "Davidson numbers":
Since is already known to be composite when n is composite (not proven in class, but it can be proven), but is not always prime when n is prime, then one concludes that prime numbers are not distributed as abundantly among "Davidson numbers" as among natural numbers. As a small illustration, consider that there are 14 prime numbers up to 43:
But there are only three prime numbers among the corresponding "Davidson numbers":
This result should not be surprising, however, considering that the size of "Davidson numbers" grows much faster than the natural numbers, nearly geometrically, in fact. Since primes are scarcer amongst larger numbers it may be that the distribution of primes among "Davidson numbers" simply reflects the relative frequency of primes among numbers of their size. For example, the probability that is prime may be roughly the same as the percentage of prime numbers in the neighborhood of .
Just how scarce are primes among the "Davidson numbers"? Hard to say. Literally. There is a known estimation for the numbers of primes less than a large number N. That estimation was a major accomplishment in number theory in the 19th Century. To quantify scarcity in light of such an estimate would mean to find a similar measurement for the distribution of primes among "Davidson numbers." It seems to be a very difficult problem and most likely an unsolved problem.
Thus ended our number theory excursion. The author had been musing about the logistics of creating a twenty tone musical scale and launched into a quick comparisons of how one might create a twenty tone chromatic scale, which appears to be nearest in mathematical structure to our familiar twelvetone scale. The name of the person who first demonstrated this fact deserves mention in this space and will be so honored as soon as the author locates his name.
For a twelve tone chromatic scale consider the sequence of numbers generated by starting with 5 and adding 7 (mod 12):
Now take the first seven of these numbers and arrange them in numerical order from low to high:
We'll consider those the white keys on our keyboard. The remaining numbers will represent the black keys. Our keyboard is labeled thus:
Coming from the additive group the next note after note 11 would be note 12, or 0. The shortest interval between two notes with the same number is called an octave. Of course, this keyboard is much more familiarly labeled thus:
The twelve note sequence {5, 0, 7, 2, 9, 4, 11, 6, 1, 8, 3, 10}, translated to the common musical note system becomes {F, C, G, D, A, E, B, F#, C#, G#, D#, A#} and is known as the circle of fifths.
Starting on C and playing the octave of white notes ascending in order {C, D, E, F, G, A, B, C} produces the C Major scale, and a familiar "do, re, mi, fa, sol, la, ti, do" routine. The five different black notes make up the pentatonic scale used in both Chinese and Scottish music.
Interestingly, a set of three nonadjacent white keys within an octave played simultaneously are either a major triad or minor triad. For example, {C, E, G} are the components of the C Major triad, or "chord," whereas {D, F, A} define the D minor triad.
(The author demonstrated these sounds on the class piano.)
Finally, it was pointed out that keyboard instruments are tuned based on the geometric sequence formed by the common ratio of the twelfth root of 2, approximately 1.05946. This system of tuning, known as welltempering, has been widespread since 1720. Johann Sebastian Bach wrote the WellTempered Clavier that year in celebration of the freedom this gave composers to write using all the notes available since prior to that some notes sounded terribly out of tune. (There is a mathematical limitation based on the fact that for all positive integers j and k. This dooms any attempt to tune all twelve notes perfectly according to natural ratios.)
Specifically, suppose that the lowest note corresponding to 0 is tuned to a frequency of . Then all other notes (numbered in order according to the keyboard pattern above, but using integers, not numbers) are tuned according to this function:
Since all octave intervals are integer multiples of 12, the frequencies of octave intervals vary according to an integer multiple of 2. This is consistent with the musical definition of octave reaching back to ancient times.
So, how to construct a 20 tone chromatic scale? One less than half of twenty is nine and one more than half is eleven. Starting at 9 and adding 11's (mod 20), we have the following sequence:
These, of course, are the elements of the additive group .
We take the first eleven notes and call them the white keys, with the other nine notes called black keys. Thus we have the following keyboard arrangement:
If we name these notes in alphabetical order beginning with A we have:
Set of nonadjacent white keys such as {A, C, E, G, I} could be used to define "major" or "minor" chords.
It would make sense to use a welltempering system to tune the notes of the twenty tone chromatic scale according to a common ratio of the twentieth root of 2:
What would this note system sound like? Hard to say, but since clashing of tones which do not form ratios of frequencies very close to natural tunings is inevitable, it should be a given that music performed in this system will sound harsher or more dissonant than music performed in the familar twelve tone system in common use.
Returning to the twelve note chromatic scale, there is an interesting connection between it and a different mathematical group, known as the dihedral group of order twentyfour, or . Arrange the chromatic scale on a 12 sided polygon (or 12gon) like so:
Let's define to be a counterclockwise rotation of 30 degrees, or one note. Then is a rotation of two notes, or is a rotation of k notes. Suppose we take a note, say E, and let act on it. Then (E) = C#. You could call a "transposition of three halftones down," terminology that would resonate with a musician.
The collection of possible rotations would be the collection of powers of . Since twelve of these rotations, or would bring you back to the starting point, or the identity, they comprise the cyclic group of order 12 and thus are isomorphic to the additive group .
Now let's consider a reflection . By this we mean "flipping" the circular arrangement of notes over along some arbitrarily chosen axis. Suppose we choose the axis to be the line connecting C and F#:
Then acting on the identity arrangement give this result:
One well imagines that a second reflection would restore the arrangement to the original, so that is the identity.
By combining the twelve rotations with the two reflectionsessentially reflection or no reflectionthere are twentyfour different such 12gons of notes, thus the term dihedral group of order twentyfour. This group, , is nonabelian, which means noncommutative. (Noncommutative menas simply that some multiplications, not necessarily all, do not commute.) It could easily be verified, for example, that .
Now here's a curiosity. If you take an arrangement of notes called a major triad, say, {C, E, G}, and rotate them on the 12gon in any way, you'll end up with another major triad. The same goes for a minor triad, say, {D, F, A}. But if you reflect these triads (by applying ) they change character; major triads become minor and vice versa. (This was demonstrated on the class piano. The mathematical language and proof for this phenomenon was developed and presented in a talk given by the author at the Univ. of So. Calif. in 1986.)
With the remaining time the class now started on a completely new topic, sphere packing in two dimensions.
How many objects of a certain size and shape can be packed into