Looking for solutions

by Rebecca Priestley / 28 March, 2013
Would you like to win $1 million? Just solve a key mathematical problem.
Looking for Solutions
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Fermat’s last theorem remained unproven for more than 300 years. Are there other equations that remain to be solved or proven?
– Dr Terry Jones

Answer: Pierre de Fermat was a French lawyer and mathematician who lived in the 17th century. While reading a translation of an Ancient Greek text, Arithmetica, Fermat wrote a note in the margin, next to the Pythagorean equation k2 = u2 + v2. He wrote that he had discovered a “truly marvellous proof” that it was impossible to solve this equation for any power higher than 2, that is: cn = an + bn was unsolvable if n > 2. That was in 1637. For more than 300 years, mathematicians tried to prove what became known as Fermat’s last theorem. Some mathematicians managed to prove the equation for specific numbers, but it was not until 1995 that British mathematician Andrew Wiles provided full proof of Fermat’s last theorem.

That was just one of many high-profile mathematical problems solved last century, says Geoff Whittle, a professor of mathematics at Victoria University of Wellington. One of the most prominent problems when Whittle was a student was what is now called the four colour theorem.

“If I give you a map of the United States, and ask you to colour in all the states without giving the same colour to any two states with a common border, how many colours do you need? People had thought about this problem for about 100 years, then in 1976, Kenneth Appel and Wolfgang Haken provided an extraordinarily long and complicated proof that the answer is four. When I first saw that, it blew me away. I thought surely that’s not possible? But you can take any map, and four colours, and you can always colour it in so that no two adjacent countries share a colour.

“There are many, many unsolved problems in mathematics,” says Whittle. “Indeed, professional mathematicians see their world as a vast continent of unexplored territory, with only a few small settlements of known mathematics.”

In 2000, the Clay Mathematics Institute stated seven key mathematical problems for which it would offer a prize of US$1 million if they were solved. Poincaré’s conjecture – a topology problem posed by French mathematician Henri Poincaré in 1904 – was solved by Grigori Perelman in 2006. One problem that particularly interests Whittle, the so-called P vs NP problem, is also on the list.

“Here’s a problem that we would all like to solve,” says Whittle. “Say I want to visit 300 cities and want to find the cheapest itinerary to get me to all of the cities. I can write a computer program, and that program will work, but it will take the computer about 2300 steps. That’s a significantly bigger number than there are atoms in the universe. No computer can do that.

“Almost all mathematicians believe there’s no way we can program a computer to solve this problem quickly. But we cannot yet prove that an efficient algorithm doesn’t exist. This is not just interesting from a historical point of view, it’s also fundamental, because this ability to decide whether or not there’s a way you can program a computer to solve a problem quickly is something we’d like to know.”

Clay Mathematics Institute Millennium, Prize Problems, www.claymath.org/millennium/

10x10 lecture series

Mathematicians play a central role in the drive to understand and deal with many of the challenges facing our planet and society, says the Royal Society of New Zealand. In a nationwide lecture series, with 10 speakers over 10 locations over 10 months, mathematicians will speak about the application of mathematics to problems in climate prediction, finance, forensics and more.

Full details at www.royalsociety.org.nz/10x10.

Send questions to science@listener.co.nz


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