The Talking Stick Blog

News, Updates, Program Recaps, and Homeschooling Information

Fermat 1 and 2: From Trial and Error into Proof

(October 31 and November 7, 2016)

“What are the greatest human accomplishments in history?” I asked at session one of our teen math circle. The students generated a list on the board. “What made them great accomplishments?” I asked. The students gave reasons for the greatness of each accomplishment listed. We did the same for a narrower version of the same question: “What are the greatest accomplishments of the human mind?”

We were quickly analyzing this question through mathematical eyes – questioning and defining terms, noticing and stating assumptions:
• Does it have to be necessary to be great?
• Does it have to have a purpose to be great?
• Does it have to be an accomplishment of the mind to be great?
• What does necessary mean?
• Does necessary apply to humanity or the individual?
• Does necessity depend upon one’s circumstances? (i.e. Is repairing a broken arm necessary if you’re a major league baseball player? A farmer? Etc.)
• Is it “necessary” if it’s only necessary for progress/growth but not survival?

The students come to consensus on stating the assumption that necessity includes accomplishments that engender progress and that are relevant to all of humanity. Then they had some disagreement about the definition of progress (is that an absolute?), productivity, and purpose. They used the example of music to frame the debate.

DELVING INTO MATHEMATICS

“What are the greatest mathematical accomplishments in history?” I then asked. The group didn’t know of many, but made the distinction between math as an art versus a tool. They decided to include both realms in their definition of great mathematical accomplishments:
• Archimedes (the math behind volume and leverage)
• Euclid (his work on collecting and organizing mathematical information)
• Cantor (set theory, infinity)
• Pythagoras (the Pythagorean theorem)

They reviewed the Pythagorean theorem – how it works and its usefulness. Then I introduced Pierre de Fermat. I told of his background as a civil servant trying to avoid the ire of Cardinal Richelieu, and a bit of the life of a math aficionado in those times. Then I introduced the problem that Fermat is famous for:

FERMAT’S LAST THEOREM

For the equation a^n + b^n = c^n, is there any integer n>2 for which a, b, and c are all integers?

We talked about what is a theorem, what was Fermat’s answer to the question above, and why people call it his “last theorem.” The students hypothesized about why Fermat wanted to share his results without sharing his proofs of those results.

Then the students decided to use inductive reasoning to develop their own conjectures. We talked about how inductive reasoning is not a logic-based approach that mathematical proof can rely on, but how it can be used to find counterexamples for conjectures. (Mathematical induction, however, is a legitimate part of the gap-free logic upon which proof depends. We didn’t talk about induction in this session, but we will.)

They acknowledged that it’s easy to come up with numbers that work for n=1 and n=2. So they decided to test numbers by stating n=3. They generated a list of perfect cubes up through 15 cubed and tried to combine them in a way that would yield a sum that was also a perfect cube. They noticed that when a=7 and b=10 the sum is close to a perfect cube. And when a=6 and b=8 the sum is only one unit away. They wanted to see how close they could get when they increased to n=4. They found that when a=6 and b=7 the sum is a few hundred away from a perfect cube.

TIME FOR A CONJECTURE

They got excited at a seeming pattern. They posited the conjecture that the larger the n, the further away an + bn is from an integer that is the nth power of an integer. There was a bit of a struggle to verbalize that conjecture, demonstrating the need of precision of language in mathematics. (If you can’t communicate your conjectures to others, what’s the point?) The students tested the conjecture for a while, but finally rejected it with a counterexample.

“Are we going to be able to do the Fermat proof by the end of this course?” asked R.

“No,” I said. “But you will have a good idea of the history and philosophy involved in the story, and an understanding of some of the mathematical methods and concepts people used to attack the problem over the centuries. Very few people in the world can understand the proof.” The students seemed satisfied, although of course they’d love to understand the proof of this seemingly simple problem.

MOVING FROM TRIAL AND ERROR INTO PROOF

In week 2, I asked the students to solve the problem of The Mutilated Chessboard. They worked using trial and error for quite some time and suspected it might be impossible, but didn’t know for sure. They used the “changing the problem” tactic to gain more insight, but still lacked certainty. We then worked together on a mathematical solution that offered certainty. We then worked together on an algebraic proof of the Pythagorean theorem and two different proofs of the infinite number of Pythagorean Triplets. If we want to prove Fermat’s Last Theorem, we need better methods than trial and error so now we’re developing those methods.

“What else is Fermat known for, besides his last theorem?” asked W. I told them the storied history of Friendly Numbers, which may have been Fermat’s first brush with mathematical celebrity. This is discussed nicely in the book we’re using for most of the course material, Fermat’s Enigma by Simon Singh.

We then ended the session with discussion on why the proof of Fermat’s Last Theorem is considered such a great accomplishment.

To view the full photo album from these sessions, click here.