A number theory story

A friend asked me to investigate how to write algorithmically generated fraction addition problems where the sum would always be reducible or irreducible. Why? It seems unfair, in a testing situation, to have one addition problem lead to a reducible sum while another is irreducible.

I remembered thinking about this problem years ago when I first taught arithmetic to adults at a community college. I recall experimenting to determine the proportion of sums that would be reducible. At about that time I was also interested in this new fad, the World Wide Web. Open access to information was all the buzz. Online privacy and freedom were also concerns, and still are. A guy named Zimmerman was being sued by the federal government about releasing software that would allow one to encrypt their personal internet communications. The encryption algorithm of the day was RSA, which involved factoring LARGE prime numbers. Although I had seen a little number theory in a history of math class as a student, these two events were my introduction to number theory.

About 10 years later, I studied number theory for a summer, mainly focusing on Euler’s Totient function, counting how many numbers are less than and relatively prime to a given number. I can’t remember why I was studying this, maybe fractions were a motivation since I was teaching arithmetic then. More likely, is that a friend said, “hey Gary, check this out.”

This time, December 2016, a quick internet search revealed a theorem with exactly the result I was interested in, a rule based on the prime factorizations of the denominators to determine when a pair of denominators yields a reducible sum. Although this allowed me to write a few algorithms to my colleague’s specifications, it revealed many more questions. Documenting my algorithms lead to interesting patterns.

Let’s engineer a sum that will reduce.

Pick an odd number. OK, 3 is good.

Now pick another. 17, good.

Let’s double each of those and use 6 and 34 as denominators.

Next we need to pick numerators so that the fractions don’t reduce.

How about `1/6` and `5/34`.

The lowest common denominator would be `2*3*17=102`

`1/6+5/34=(1*17)/(6*17)+(5*3)/(34*3)=(17+15)/102=32/102`

This sum reduces from `32/102` to `16/51`.

They keys to my trick here are that I had denominators for which 21 is the greatest even common factor and that the numerators didn’t cause either fraction to reduce.

OUTLINE

  • History?
  • Play with a few sums, always reducible, never, and sometimes.
  • State the theorem.
  • State some questions that have arisen
    • Why is the table of sums for two denominators symmetric?
    • Why must any reducible sum, reduce by a factor of the gcd of the denominators?
    • Is the reducible pattern in each row part of the same pattern with different starting points?
    • Why is the number of reducible sums the same in each row?
    • Is there an algorithm for picking numerators for the conditional cases?
    • What is the percent of irreducible sums of reduced fractions? All fractions?
  • Show the explorations tools
  • State some false conjectures
  • State some true conjectures
  • What is next?

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