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


Question:
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

Latest

Lecretia Seales' widower makes his case for death with dignity
103655 2019-03-20 00:00:00Z Health

Lecretia Seales' widower makes his case for death…

by Matt Vickers

Ahead of a report back on the End of Life Choice Bill, Matt Vickers, widower of assisted dying advocate Lecretia Seales makes his case.

Read more
Sid at the French Café to host a special lunch to support Christchurch victims
103751 2019-03-20 00:00:00Z Auckland Eats

Sid at the French Café to host a special lunch to…

by Metro

A special luncheon will be held to raise money in support of the victims and families affected by the Christchurch mosque shooting.

Read more
Tech giants accused of inciting violence for hosting mosque shooter's livestream
103603 2019-03-19 10:23:15Z Tech

Tech giants accused of inciting violence for hosti…

by Vomle Springford

Facebook, Google and other tech platforms are being condemned for hosting the Christchurch shooter's video.

Read more
'We've failed miserably': Critics condemn spy agencies' surveillance strategy
103607 2019-03-19 00:00:00Z Crime

'We've failed miserably': Critics condemn spy agen…

by Phil Pennington

Former Race Relations Commissioner Dame Susan Devoy said the response to Muslims over what they see as a growing threat to them was "diabolical".

Read more
Party leaders saying little on gun law reform
103601 2019-03-19 00:00:00Z Politics

Party leaders saying little on gun law reform

by Gia Garrick

Governing parties are giving no clues on how the country's gun laws will change following Friday's terror attack in Christchurch.

Read more
Test of faith: Does New Zealand have anything to fear from radical Islam?
70722 2019-03-19 00:00:00Z Social issues

Test of faith: Does New Zealand have anything to f…

by Joanna Wane

Almost three years ago, the Muslim community in Auckland welcomed me into their world with warmth, trust and open arms.

Read more
Saziah Bashir: 4 things you should do following the Christchurch terror attack
103634 2019-03-19 00:00:00Z Social issues

Saziah Bashir: 4 things you should do following th…

by Saziah Bashir

What can we do? Where to from here? People have to recognise the Muslim community is grieving.

Read more
Christchurch shooting: How the world is reacting to the terror attack
103651 2019-03-19 00:00:00Z World

Christchurch shooting: How the world is reacting t…

by Ryan J Holder

In the wake of the Christchurch terror attack, there is grief, despair, anger and a righteous sense that things need to change.

Read more