for n = 1, 2, 3, . . . 10. As in the case for repetends of length
5 given on day 3, the prime factors of numbers consisting
of a string of n 1's are the potential primes whose reciprocals have
repetends of length n.
The challenge from the previous class was the following:
Class members had factored numbers of the form
for n = 1, 2, 3, . . . 10. As in the case for repetends of length
5 given on day 3, the prime factors of numbers consisting
of a string of n 1's are the potential primes whose reciprocals have
repetends of length n.
The class produced the following chart:
A lively discussion ensued on how best to determine if 333,667 is prime or composite. Since the square root of this number is around 577, a prime factor search would require dividing 333,667 by all primes up to 577. One student had a list of primes up to 10,000 he had found on the internet, and began methodically dividing 333,667. George Carver, who seems particularly skilled at internet searches (this is a guy who could track down the name of the host of the ancient TV show "Queen for a Day" in under a minute!) quickly found a web site which checks for primality and verified that 333,667 is prime.
The author worked out a calculator program on his TI-81 for the class that will determine the smallest prime factor of a number by an inefficient search through all possible odd number divisors. This should work on all TI graphing calculators:
:1->P
:Input N
:
N->L
:Lbl A
:P+2->P
:If P>L
:Goto C
:N/P->X
:If (X-IPart X)>0.000000001
:Goto A
:Lbl B
:Disp P
:End
:Lbl C
:Disp N
The program asks for an input value, the natural number to be factored. If it returns a smaller value then that is the smallest factor. If it returns the original number, then that number is prime.
The calculator program above takes about seven seconds to determine that 333,667 is prime. This is very slow compared to a desktop computer, but far faster than manually dividing prime by prime on a calculator!
The author related to the class about his experience trying to determine whether 1,111,111,111,111,111,111 (a string of nineteen 1's) is prime during the summer of 1992 by using an already out-dated 8086 IBM computer and a program he wrote in BASIC. It took several weeks, perhaps thirty hours of computing time. Using Maple V, a mathematics program produced by Waterloo Maple, on a much faster personal computer made in late 1997 (Mac PowerBook 3400c, 200 MHz), this number was now determined to be prime in about a second.
Let's define 
, the number consisting of a string
of n ones. ("d" for "Davidson numbers")
Then 
= 1,111,111,111,111,111,111, and , as noted, is prime. Then
the equation
is equivalent to the equation
,which means that p \ 9,999,999,999,999,999,999.
But the only prime divisors of 9,999,999,999,999,999,999 are 3 and
. Thus is the only prime number whose reciprocal has a repetend of length
of 19.
The class examined other numbers of the type 
. It seems that when n is a composite number, then must also be
composite. (That can be proven.) Consider some examples:
In fact, if 
then it appears
that both 
and 
.
But when n is prime, is not necessarily
prime. In fact, the next prime after n = 2, or 
, is . It appears, then, that prime numbers are scarce in the set
of numbers of the form , which means they appear to be distributed
less frequently than within the set of integers. Consider a one to one correspondance
between the set of natural numbers beginning with n = 2 and the set
of numbers of the form beginning with n = 2. Every composite
number in the first set corresponds to a composite number in the second
set, but not every prime number in the first set corresponds to a prime
in the second set.
The question was raised about the length of repetends for
1 ÷ n when n is composite. If we define a function
such that represents
the length of the repetend of 1 ÷ n, a cursory examination
of several examples indicated the following conjecture:
Conjecture: If n is a composite integer such that
where the p's are prime components of n, then
.
In other words, the length of the repetend of 1 ÷ n is the product of the lengths of the repetends of the reciprocals of its prime factors.
A conjecture is a hypothesis, or unproved theorem. The issue above is an interesting one that we did not pursue, but later an example examined by Carver indicated the conjecture needs refinement, at least for powers of prime factors:
, but
George Carver gave a quick explanation of the permutation
group of three objects using a triangle which can be rotated or reflected,
sometimes known as 
. (Also known as the dihedral group of order
six.) The class was given the challenge of completing the multiplication
table for this non-abelian (non-commutative) group.
Thus ended our first topic of the course, which involved an excursion into number theory and groups. The class was also given the following writing assignment:
April 8, 1999
Write a two or three page paper on something you learned the first two weeks. I prefer to have it next Tuesday since we begin a different topic that day. Here are some guidelines to consider:
·Since few people possess math equation writing software and have the skill to use it, neatly handwritten is acceptable. If you prefer to type the sentences and hand write the math, that is certainly acceptable. I'm not expecting photo-ready work.
·Double space. Reading and understanding mathematical prose if difficult enough without the obstacle of cramped writing.
·Unless explaining these terms are integral to your paper, you may assume that your reading audience, if there was one, already knows what repetends, modulo arithmetic, and mathematical groups mean. You may also assume they know stuff like "cyclic group of order 12 under multiplication," but not that is a cyclic group of order twelve under multiplication unless you indicate that's what your notation means.
If you can think of something, end your paper with a topic or question for further investigation related to what you wrote about.
I will read the papers and offer constructive criticism. My biggest concern is that the mathematics you present is sound and clearly communicated. If your paper is really good and you agree to it, I'll put it on the web site.
