Fermat’s fiendishly difficult problem

Pierre de Fermat (1607-1665)

It was a famous unsolved maths problem that had stood for hundreds of years – Fermat’s Last Theorem (FLT) – then along came Andrew Wiles, who solved it in 1994. That’s the beginning and end of it really, though the details of the efforts to solve it provide a fascinating tale. The main reason is that FLT is a seemingly simple problem, easily understood by anyone with a cursory understanding of high school maths.

Most of us have encountered plenty of solutions of the equation a2 + b2 = c2, which are known as Pythagorean triples (e.g. 32 + 42 = 52 ). But for the general equation an + bn = cn , what if n is greater than 2? Fermat claimed it wasn’t true; that is, there are no positive integers (counting numbers 1, 2, 3, 4, …) greater than 2 that fit the equation.

Andrew Wiles

I was one of those high school students who came across FLT, and the added joy for me was that at that time, many moons ago, it was unsolved – Wiles had not performed his magic. And it might as well be magic, I’m afraid, as his proof is so complex that only a select group of mathematicians understand it. I have an honours degree in maths, and I am not in that group; far from it. So, many of us feel just a tad unsatisfied with the proof (as with the solution to the famous Four Colour Problem), and wonder if there just might be a more elegant and simple proof sitting out there somewhere, just waiting to be discovered.

Or rediscovered. You see, Fermat himself stated that he’d found a proof, but claimed he didn’t have room for it in the document he was annotating.

This and other historical details are included in an essay I wrote for a unit ‘The History of Mathematics’ while studying at the University of Tasmania (again, before Wiles). I’ve dug out a copy, which includes notes I’d added (and the proof for n = 4), and here it is. If it inspires you, you’re under 40, and you can solve it more elegantly than Wiles, you’ll likely be awarded a Fields Medal, the supreme maths prize which eluded him (he was over 40 years of age when he solved FLT).

When Trollope met Turgenev

One of the joys of retirement is the time for reading. My reading follows no particular path, comprising a mix of fiction and non-fiction, as I seek out writers who enhance this simple and rewarding pleasure. Authors I’ve enjoyed include David Lodge, Phillip Roth, Amis father and son, Patrick Leigh Fermour, Christopher Hitchins and Hugh Trevor-Roper, plus a few relative ‘unknowns’ who’ve told remarkable stories, such as Carmen Callil (Bad Faith) and Heather Rossiter (Lady Spy, Gentleman Explorer). Continue reading

Hello to all that

In recent years we’ve been treated (if that’s the right word) to multimedia offerings that seemingly reveal the realities of World War I. Two specific examples are the much-lauded film 1917, and, more significantly, Peter Jackson’s magnificent They Shall Not Grow Old, amassed from archival footage from the Imperial War Museum. But what about the printed form? The book that resonated with me is Robert Graves’ Goodbye to All That, which I’ve recently re-read after a couple of decades. Continue reading

When Kipling met Twain

Blessed is the man who finds no disillusion when he is brought face to face with a revered writer.

Rudyard Kipling interviewed Mark Twain in 1889. This surprising fact was revealed to me as I perused one of the books which emerged when I recently successfully ‘downsized’, moving across the road from a house to a townhouse. It was an opportunity to take stock of my motley collection of books, discarding a few (not as many as I should have), reminding myself to re-read some favourites and planning to read a few unread titles. Continue reading

Who is that on the $50 note?

Australian currency is a curious mixture, with the coins dominated by the monarch and varieties of fauna, apart from the $2 coin, which has QEII on one side and Gwoya Tjungurrayi, an Aboriginal tribal elder, on the other. The notes exhibit a motley collection of Australian notables, some more well known than others (by me, anyway: BanjoPaterson, Edith Cowan, Nellie Melba, John Monash). Less well known are Mary Reibey on the $20 note (a rags-to-riches story of a girl transported to Australia as a 13-year-old convict) and the Aboriginal man depicted on the $50 note, David Unaipon. Continue reading

Who sent the first rocket into space?

Very few people seem to know who sent the first rocket into space – well, I didn’t anyway. The following quote provides a not-very-helpful clue:

… we have invaded space with our rocket and for the first time – mark this well – have used space as a bridge between two points on the earth; we have proved rocket propulsion practicable for space travel. To land, sea and air may now be added infinite space as a medium of future inter-continental traffic. This day … is the first of a new era in transportation, that of space travel.

Who said these words, when were they spoken and what was the occasion? Continue reading

Surfing goes full circle

My first memories of being in surf are as a child in the late 1950s, floundering around at Marion Bay (Tasmania) on what we called a Li-Lo. It was longer than the ‘surf-o-planes’ (surf mats) that had been invented in Australia in the 1930s, more like a floating mattress than a surfing device. My only recollection of it is being spun sideways and thrown off as the whitewater broke over me. I have no pictures, but recall it being blue and white striped on the top and plain blue underneath. Continue reading

A reminder of my mathematical past … and my connection with a princess

Browsing 3 Quarks Daily, my eye caught a particular title: ‘AI has cracked a key mathematical puzzle for understanding our world‘. Reported in the MIT Technology Review, the article’s sub-title explains that “Partial differential equations can describe everything from planetary motion to plate tectonics, but they’re notoriously hard to solve.” Continue reading

Interactive video teaching and learning – decades ago

Students and teachers around the world have been engaged in various forms of online and video-based education this year. Overall it seems to have been reasonably successful, another step in the advances in teaching with technology.

But is it new and innovatory? No, of course not. And, coincidentally, I’ve been cleaning out cupboards, drawers and shelves lately, in preparation for a move, and came across a project report from over three and a half decades ago concerning this mode of education. Continue reading