Our maths problems don’t add up

OUR civilisation ‘rests’ on the ‘three Rs’.

Irish students excel at two, reading and writing, but are down the class for the third, arithmetic, or ‘sums’.

To prosper at science, we’ll have to pull up our numerical socks. A film with a mathematical theme was shown at the Science 2012 Festival last week. Fermat’s Room is a Spanish thriller directed by Luis Piedrahita and Rodrigo Sopeña.

Pierre Fermat, of Basque origin, was a lawyer in Toulouse in the early 17th century. Fluent in six languages, his passion was mathematics.

Fermat laid the groundwork for developments in calculus and coordinate geometry, but he’s remembered for contributions to number theory, as outlined in letters to the leading mathematicians of the day. His ‘last theorem’ has become famous.

A ‘conjecture’ is a proposition which has yet to be proved. Once proof is demonstrated, it becomes a ‘theorem’. Leafing through his father’s copy of Arithmetica, the ancient Greek maths treatise, Fermat’s son found a conjecture scribbled in a page margin.

His father had added a note saying that there wasn’t space to give the proof there and then. Nor was it to be found elsewhere in his writings. I won’t state the conjecture here; we don’t want to frighten the horses.

Mathematicians set about proving the proposition but it defied all attempts. A few thought they had cracked it, only to have their hopes dashed. The Guinness Book of Records once described it as the most difficult unsolved problem in all of mathematics.

In 1963, ten-year-old Andrew Wiles was on his way home from King’s College School, in Cambridge, when he stopped off at the local library. There, he stumbled on a book that described Fermat’s notorious conjecture. Wiles, who went on to become a mathematician, thought long and hard about the problem.

Thirty years later, he had developed what he thought was a proof and presented it at a conference in Oxford.

However, an error was found, which took him two years to get around. In 1995, Wiles published his amended proof in a special edition of the Annals of Mathematics; 358 years after it was first proposed, Fermat’s ‘last conjecture’ had become Fermat’s ‘last theorem.’ By then, it was as famous as the one attributed to Pythagoras 2,000 years previously.

The plot of Fermat’s Room concerns four characters, one of whom claims to have proved the famous Goldbach conjecture. Trapped, the four must solve a succession of problems of the ‘fox, goose and sack of meal’ variety, if they are to avoid a nasty death.

The viewer is drawn into trying to solve the riddles with them as the clock ticks. By the end of the film, he or she will have spent at least an hour engaged in mathematical gymnastics. Wiles’s achievement ruled out using the Fermat conjecture for the plot of the film. The famous name, however, appears in the title; ‘Goldbach’s Room’ would, presumably, have far less clout at the box-office.

Writing to the great Swiss mathematician Leonard Euler, in 1742, Christian Goldberg suggested that all even numbers other than two are the sum of two primes; for example, 12 = 7 + 5, 18 = 11 + 7 etc. A prime number, the cognoscenti will know, can’t be divided by another one. This innocuous-looking conjecture, as far as I know, has never been proved.

But is the low-key, low-budget film worth seeing? Although not everyone’s cup of tea, it will appeal to most viewers. Ultimately, the loose ends are neatly tied up and the plot lines transparently resolved, the elements finally falling into place like the elegant solution to a mathematical problem.

Publisher Tony Faber has offered a million dollars to the person who cracks Goldberg’s Conjecture. Do so, and you’ll restore Ireland’s battered reputation for mathematical incompetence.