The Talking Stick Blog

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FERMAT 3 and 4: Proofs Proofs and Proofs

WHAT IS THE MOST IMPORTANT NUMBER IN THE WORLD?

The students came up with a list that includes just about everything that mathematicians say is important. (Non-mathematicians have a completely different list, though!) What would a mathematician say is the most important type of number in the world? This was tougher. Lots of conjectures. I finally used some leading questions to move from the idea of atomic structure (basic particles) to the fundamental theorem of arithmetic (every number can be written as a distinct product of prime numbers). That was a big enough hint to elicit prime numbers, which play

The students came up with a list that includes just about everything that mathematicians say is important. (Non-mathematicians have a completely different list, though!) What would a mathematician say is the most important type of number in the world? This was tougher. Lots of conjectures. I finally used some leading questions to move from the idea of atomic structure (basic particles) to the fundamental theorem of arithmetic (every number can be written as a distinct product of prime numbers). That was a big enough hint to elicit prime numbers, which play huge rule in the evolution of thought on Fermat’s last theorem.

The students came up with a list that includes just about everything that mathematicians say is important. (Non-mathematicians have a completely different list, though!) What would a mathematician say is the most important type of number in the world? This was tougher. Lots of conjectures. I finally used some leading questions to move from the idea of atomic structure (basic particles) to the fundamental theorem of arithmetic (every number can be written as a distinct product of prime numbers). That was a big enough hint to elicit prime numbers, which play huge rule in the evolution of thought on Fermat’s last theorem.

SYMBOLIC REPRESENTATION IN PROOFS

Are there any integers that are exactly one unit more than a perfect square and one unit below a perfect cube? With some work, the students found that the number 26 meets this requirement. Do any others? We talked about Fermat’s work on this question.* We explored this problem with numbers, number lines, and words. Then I challenged the students to express the problem with algebraic and logical symbols. They heroically rose to the occasion.

Are there any integers that are exactly one unit more than a perfect square and one unit below a perfect cube? With some work, the students found that the number 26 meets this requirement. Do any others? We talked about Fermat’s work on this question.* We explored this problem with numbers, number lines, and words. Then I challenged the students to express the problem with algebraic and logical symbols. They heroically rose to the occasion.

WHAT IS DESIRABLE IN A PROOF?

What makes one proof preferable to another? The students devised their own proof that there is an infinite number of Pythagorean triplets. Then I showed them Euclid’s proof. We discussed characteristics of proofs. What makes a proof elegant? The students decided that while Euclid’s proof is cool and clever, theirs was more direct.

What makes one proof preferable to another? The students devised their own proof that there is an infinite number of Pythagorean triplets. Then I showed them Euclid’s proof. We discussed characteristics of proofs. What makes a proof elegant? The students decided that while Euclid’s proof is cool and clever, theirs was more direct.

What makes one proof preferable to another? The students devised their own proof that there is an infinite number of Pythagorean triplets. Then I showed them Euclid’s proof. We discussed characteristics of proofs. What makes a proof elegant? The students decided that while Euclid’s proof is cool and clever, theirs was more direct.

VISUAL PROOFS

We did a few visual proofs that

We did a few visual proofs that a x b = b x a. What was challenging for the students was to do it without numbers or calculations, but they did it.

PROOF BY CONTRADICTION

Here’s another important tool in work on Fermat’s last theorem: you attempt to prove the opposite then find a contradiction. The students and I worked through Euclid’s proof that the square root of two is irrational. This proof is a specific type of proof by contradiction called “infinite descent,” later used heavily by Fermat.

Here’s another important tool in work on Fermat’s last theorem: you attempt to prove the opposite then find a contradiction. The students and I worked through Euclid’s proof that the square root of two is irrational. This proof is a specific type of proof by contradiction called “infinite descent,” later used heavily by Fermat.

HOW DO YOU NAME A THEOREM?

One thing that has been fun in our course so far is laughing about how Fermat would publicly announce that he had proved some interesting thing, but then refuse to share the actual proof. He shared enough for people to know that he really had the goods, but how did he do it? Here’s an example: all prime numbers can be put into two categories – those that can be expressed in the form 4n+1 (i.e. 13, which is 4x3+1) or those that can be expressed in the form 4n-1 (i.e. 11, which is 4x3-1). Fermat claimed that he could prove that all those primes in the 4n+1 category could also be expressed as the sum of two squares (i.e. 13 = 9 + 4). But did he share his proof? Of course not. The rock-star mathematician Leonhard Euler did proof it about a century later. So now that the conjecture had a proof, it was no longer a conjecture but a theorem. Theorems typically have names. So what to call it? The students came up with a bunch, which were all similar to the four different names of this theorem that I’ve been able to find so far.

One thing that has been fun in our course so far is laughing about how Fermat would publicly announce that he had proved some interesting thing, but then refuse to share the actual proof. He shared enough for people to know that he really had the goods, but how did he do it? Here’s an example: all prime numbers can be put into two categories – those that can be expressed in the form 4n+1 (i.e. 13, which is 4x3+1) or those that can be expressed in the form 4n-1 (i.e. 11, which is 4x3-1). Fermat claimed that he could prove that all those primes in the 4n+1 category could also be expressed as the sum of two squares (i.e. 13 = 9 + 4). But did he share his proof? Of course not. The rock-star mathematician Leonhard Euler did proof it about a century later. So now that the conjecture had a proof, it was no longer a conjecture but a theorem. Theorems typically have names. So what to call it? The students came up with a bunch, which were all similar to the four different names of this theorem that I’ve been able to find so far.

PROOFS BY INDUCTION

If you set out to prove Fermat’s last theorem (which used to be called a conjecture), you’ll want to have proof by induction in your toolbox. This is an algebraic tool where you find one case (number) where your conjecture works, then assume that it works for some other number k, then prove that it also works for k+1, then make a logically sound generalization to all numbers. The students worked through one such proof: that the sum of the first n natural numbers equals (1/2)(n)(n+1).

If you set out to prove Fermat’s last theorem (which used to be called a conjecture), you’ll want to have proof by induction in your toolbox. This is an algebraic tool where you find one case (number) where your conjecture works, then assume that it works for some other number k, then prove that it also works for k+1, then make a logically sound generalization to all numbers. The students worked through one such proof: that the sum of the first n natural numbers equals (1/2)(n)(n+1).

And that was the end of our third session. Week four began with me asking this question: how could you prove that the sum of the first n positive odd integers = n squared? I mentioned that there is a visual proof that I wanted to show them, but the students wanted to attempt to prove it by induction. So they did! Exclamation point because this proof is a little more complicated than last week’s induction proof because now we’re limited just to odd numbers, and that has to be accounted for algebraically. Also exclamation point because the students realized this before I did. Then I showed them the visual proof too.

NUMBER THEORY HISTORY

In what realm of mathematics do you think Fermat’s last theorem lies? The students thought it was both algebra and number theory. We talked about the history of numbers, specifically what kind of things in history or math would have led people to discovery/invention of certain kinds of numbers – the natural (counting) number? negatives? fractions? irrationals? imaginary numbers? We also did some dramatizations that make clear the need for numbers in general. Our talk of ancient Egyptian rope stretchers (land surveyors) led to a detour from questions relevant to Fermat: if you had a rope of length x and used it to measure off a parcel of land, would the shape of the land area matter if you wanted to maximize its area? If so, which shape would maximize it? The students formed a conjecture then tested it.

In what realm of mathematics do you think Fermat’s last theorem lies? The students thought it was both algebra and number theory. We talked about the history of numbers, specifically what kind of things in history or math would have led people to discovery/invention of certain kinds of numbers – the natural (counting) number? negatives? fractions? irrationals? imaginary numbers? We also did some dramatizations that make clear the need for numbers in general. Our talk of ancient Egyptian rope stretchers (land surveyors) led to a detour from questions relevant to Fermat: if you had a rope of length x and used it to measure off a parcel of land, would the shape of the land area matter if you wanted to maximize its area? If so, which shape would maximize it? The students formed a conjecture then tested it.

GENERALIZING TO ALL CASES OF A PROBLEM

Since Euler contributed significantly to Fermat’s last theorem, I shared another of Euler’s famous work: the Konigsburg Bridge Problem. The students spent a good chunk of class time working on this, and finally came up with a solution. I asked: is it enough to just answer for this case? Or should you have a method that you can apply to any situation of bridges and land masses? They worked harder. How can you come up with a solution that you can generalize to all problems in this class of problems? The students stated a conjecture that was really on target, but then spent a long time off on an unproductive tangent. With a few minutes left, I directed them back to their previous conjecture and showed them how Euler had this same conjecture and used it to basically invent (discover?) the huge area of mathematics called graph theory.

I wrapped things up with a promise to spend our last 2 sessions directly attacking Fermat’s last theorem with the tools we have.

Rodi (November 15 and 22, 2016)
*Once again, all historical anecdotes are from Simon Singh’s Fermat’s Enigma.