The Greatest Unsolved Problems In Mathematics and The Prizes For Solving Them Part 1

1st November 2016

Many people don’t realise that mathematics is a dynamic and fluid field which is constantly developing. There are disciplines in mathematics that didn’t exist 10 years ago, and new branches emerge all the time, with new discoveries, theorems and problems evolving.

People also often mistakenly think that there can’t be many unsolved problems around because since the ‘basics’ of mathematics remain static they assume that it’s just a case of pointing a mathematician at the problem and waiting.

As long as students believe this, it is hard to motivate them to explore problems and think about them in a creative way because they seems like dead end exercises.

But there are actually thousands of unsolved problems in mathematics and this is part of the joy of the subject, there is always more to discover. Some of these problems are so inherently ‘simple’ that even a child could attempt to solve them while others are so complex only mathematicians can understand them – but they still can’t solve them.

The Classic Example

You have probably heard of Fermat’s Last Theorem - we featured a report on the prize awarded for it's solution earlier in the year. Fermat's Last Theorem states that xn + yn = zn has no non-zero integer solutions for x, y and z when n > 2.

Mathematicians struggled for 357 years to solve it until British mathematician Andrew Wiles proved the theorem in 1995.

Wiles used ‘new maths’, developed by building upon the previous work of his contemporaries, with fresh insight into modern ideas. His proof relied upon many of the greatest ideas of the last 40 years plus 8 years of calculation but Wiles had been driven by the problem since he was 10.

Andrew Wiles in 1995

His success is a perfect example of how the mystery of unsolved mathematical problems drives innovation and creativity, and spurs on the development of new ideas and discoveries.

Financial Reward

Unsolved problems can also carry the additional incentive of financial rewards beyond the intellectual satisfaction and recognition. Wiles received the Wolfskehl prize for proving Fermat’s Last Theorem which was worth approximately £30,000 but in 2016 he was also awarded the Abel prize for the same accomplishment, worth an impressive £495,000!

There are actually several other unsolved problems which, if solved, would be worth a lot of money. Texan billionaire banker and number theorist D. Andrew Beal created the Beal’s Conjecture number theory theorem inspired by Fermat’s Last Theorem, stating that:

“If Ax + By = Cz, where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common prime factor.”

Beal and the Beal Conjecture

The mathematical community was impressed and in 1997 Beal offered $5,000 to the person who could prove or disprove it; he has since raised the reward to $1,000,000 as no one has managed to yet!

Beal says the prize is also meant to inspire young people to get involved with maths.

Beal’s independent prize now matches 7 of the biggest, most maddening and potentially most lucrative problems in mathematics – the Millenium Prize Problems. In 2000, the Clay Mathematics Institute announced that it would award $1,000,000 to anyone who could solve any of these 7 unsolved problems and so far only one has been - the Poincaré conjecture.

Named for French mathematician Henri Poincaré, it involves a complex problem in the field of topology - the study of the enduring properties of objects which are stretched or deformed, but not torn or reconstituted.

Poincare postulated the conjecture in 1904 and it was nearly 100 years before the reclusive, brilliant Russian Grigori Perelman posted his solution online, avoiding standard peer review formalities.

The mathematical community spent 2 years verifying his solution and when they did he was awarded the Field’s medal which he promptly refused.

Field's Medal

Enough Already

He is quoted as saying that he ‘had enough already’, despite being somewhat impoverished and living with his mother in a small flat outside St Petersburg "I'm not interested in money or fame," he said "I don't want to be on display like an animal in a zoo. I'm not a hero of mathematics. I'm not even that successful; that is why I don't want to have everybody looking at me."

Perelman is one of those mathematicians who truly embodies the wild genius man stereotype, he is a unique individual who had had a true passion for mathematics and little interest in much else, practically working in isolation on the problem for 7 years straight!

When the Clay Institute tried to award Perelman $1,000,000 he refused again and retired from formal mathematics.

He apparently felt that another mathematician, Richard Hamilton deserved the same credit, as Hamilton’s work laid the groundwork for Perelman’s breakthrough. Friends described him as impeccably honest and fair, and he felt the decision was unjust therefore he longer wanted to be associated with the formal mathematical community.

Many in the mathematical community respect and support his decision and the story has a quasi-happy ending. In 2011 the Shaw Prize Foundation in Hong Kong recognized Hamilton’s contribution to the proof of the Poincare conjecture, awarding him $500,000 which would have been half of the Clay Foundation prize.

Richard Hamilton receiving his prize in Hong Kong

So that’s alright then.

Next month we will be looking in less detail at the remaining six legendary Millennium Prize Problems, as well as some lesser known fun ones for you to think about.