3a+5b Proofs

Written by: Dr. Julian Fleron

Asked to determine all possible values generated by the Diophantine equation $3a+5b$ when $a,b ≥ 0$, students discovered their first proofs involving the infinite. The diversity of entirely different proofs was both a challenge to the teacher and a great affirmation of the importance of inquiry-based learning.

Proof 1 is what I anticipated and had in my head as I planned for our investigation of this problem. I knew that it would take some exploration, experimentation, data collection and organization for this approach to germinate into a real idea. I let that happen. I noticed very few groups heading a direction that I thought would get them to this point. Not knowing what to do without giving the entire secret of this proof away, I simply let them explore. As they did, I inquired about their strategies, data, and ideas. As I moved from table to table over the course of two 50 minute class periods I was continually challenged to interpret entirely different approaches to the problem.

Proof 2 was discovered early in this process by one group. My conversation with this group left me befuddled. They were explaining to me that it was much like the finger game Chopsticks. I had no idea what this game was or what they were trying to communicate to me. With patience, what I learned to see, and what is seen from the way they articulated their proof, is that there approach is a complexified version of the inductive approach to Chisenbop for counting by nines. This group made a model in which they are repeatedly exchanging 3s or 5s as if they are coins, in each case figuring out how to increase the total generated by $3a+5b$ by $1$. In this way they can generate all possible outputs. It is really a wonderful approach.

Proof 3 is similar to Proof 2, although here we see a group that is thinking very algorithmically. They even use the word "algorithm".

Proof 4 separates the outputs into mod 3 congruence classes. Wonderful.

Other proofs include: a separation into mod 5 congruence classes and a beautiful array in which the axes represent the values of a and b and one sees all sorts of beautiful trajectories along which all possible outputs are generated.

The following videos shows proofs from a different class (Prof. von Renesse) in which a and b could be negative numbers.