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
Photo/Getty Images

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
MostReadArticlesCollectionWidget - Most Read - Used in articles
AdvertModule - Advert - M-Rec / Halfpage


Vincent O’Malley: Why we need to open up about past Māori and Pākehā conflict
106234 2019-05-26 00:00:00Z History

Vincent O’Malley: Why we need to open up about pas…

by Sally Blundell

Calls are growing for us to take a more honest look at our past, particularly the wars over land and power that shaped the country.

Read more
Scott Morrison: How a 'doomed' PM stormed the country with one killer line
106291 2019-05-26 00:00:00Z World

Scott Morrison: How a 'doomed' PM stormed the coun…

by Bernard Lagan

As Australia’s tourism tsar 13 years ago, Scott Morrison oversaw the rollicking “So where the bloody hell are you?’’ ad campaign.

Read more
What you need to know about knee replacements
105774 2019-05-26 00:00:00Z Health

What you need to know about knee replacements

by Ruth Nichol

Replacement knee joints are giving thousands of Kiwis decades of service, but don’t rush to get one.

Read more
How a hit romcom took indigenous Aussie star Miranda Tapsell back to her roots
106072 2019-05-25 00:00:00Z Movies

How a hit romcom took indigenous Aussie star Miran…

by Russell Baillie

Miranda Tapsell tells Russell Baillie how she came up with Top End Wedding and why its Northern Territory setting means so much.

Read more
The link between cardiovascular health and dementia
105915 2019-05-25 00:00:00Z Health

The link between cardiovascular health and dementi…

by Nicky Pellegrino

New research into the brain has found that cardiovascular ill health is linked to cognitive decline and dementia.

Read more
Following the call of New Zealand's abandoned freezing works
106317 2019-05-25 00:00:00Z Life in NZ

Following the call of New Zealand's abandoned free…

by John Summers

John Summers wonders if his abiding interest in New Zealand’s abandoned freezing works is actually a long farewell to his grandfather.

Read more
Tech Week: Time to celebrate Aotearoa’s own overlooked moonshot
106359 2019-05-25 00:00:00Z Tech

Tech Week: Time to celebrate Aotearoa’s own overlo…

by Peter Griffin

“We bow down to this idea of Elon Musk or Jeff Bezos going to Mars, when here in our own country, we had the equivalent."

Read more
Kiwi composer John Rimmer: An instrumental figure
106331 2019-05-24 11:09:35Z Music

Kiwi composer John Rimmer: An instrumental figure

by Elizabeth Kerr

Contemporaries and students are paying tribute to composer John Rimmer and his musical legacy.

Read more