MitPY 7: Diophantine Teen Fans

This MitPY is a request from frequent commenter, Red Five:

I’d like to ask what others think of teaching (mostly linear) Diophantine equations in early secondary school. They are nowhere in the curriculum but seem to be everywhere in competitions, including the AMC junior papers on occasion. I don’t see any reason to not teach them (even as an extension idea) but others may have some insights into why it won’t work.

Fibonacci Numbers to the Rescue

No, really. This time it’s true. Just wait.

Our favourite mathematics populariser at the moment is Evelyn Lamb. Lamb’s YouTube videos are great because they don’t exist. Evelyn Lamb is a writer. (That is not Lamb in the photo above. We’ll get there.)

It is notoriously difficult to write good popular mathematics (whatever one might mean by “popular”). It is very easy to drown a mathematics story in equations and technical details. But, in trying to avoid that error, the temptation then is to cheat and to settle for half-truths, or to give up entirely and write maths-free fluff. And then there’s the writing, which must be engaging and crystal clear. There are very few people who know enough of mathematics and non-mathematicians and words, and who are willing to sweat sufficiently over the details, to make it all work.

Of course the all-time master of popular mathematics was Martin Gardner, who wrote the Mathematical Games column in Scientific American for approximately three hundred years. Gardner is responsible for inspiring more teenagers to become mathematicians than anyone else, by an order of magnitude. If you don’t know of Martin Gardner then stop reading and go buy this book. Now!

Evelyn Lamb is not Martin Gardner. No one is. But she is very good. Lamb writes the mathematics blog Roots of Unity for Scientific American, and her posts are often surprising, always interesting, and very well written.

That is all by way of introduction to a lovely post that Lamb wrote last week in honour of Julia Robinson, who would have turned 100 on December 8. That is Robinson in the photo above. Robinson’s is one of the great, and very sad, stories of 20th century mathematics.

Robinson worked on Diophantine equations, polynomial equations with integer coefficients and where we’re hunting for integer solutions. So, for example, the equation x2 + y2 = z2 is Diophantine with the integer solution (3,4,5), as well as many others. By contrast, the Diophantine equation x2 + y2 = 3 clearly has no integer solutions.

Robinson did groundbreaking work on Hilbert’s 10th problem, which asks if there exists an algorithm to determine whether a Diophantine equation has (integer) solutions. Robinson was unable to solve Hilbert’s problem. In 1970, however, building on the work of Robinson and her collaborators, the Russian mathematician Yuri Matiyasevich was able solve the problem in the negative: no such algorithm exists. And the magic key that allowed Matiyasevich to complete Robinson’s work was … wait for it … Fibonacci numbers.

Label the Fibonacci numbers as follows:

F1 =1, F2 = 1, F3 = 2, F4 = 3, F5 = 5, F6 = 8, …

It turns out that with this labelling the Fibonacci numbers have the following weird property:

If Fn2 divides Fm then Fn divides m.

You can check what this is saying with n = 3 and m = 6. (We haven’t been able to find a proof online to which to link.) How does that help solve Hilbert’s problem? Read Lamb’s post, and her more bio-ish article in Science News, and find out.