Fun with NUMBERS
In 1900, a mathematician called David Hilbert (1862-1943) gave a famous speech in Paris. In it, he outlined 23 open problems in mathematics that proved influential and inspirational across the whole of the last 21st century.
Many of Hilbert's problems have now been solved, but some remain the focus of research in universities today.
Inspired by Hilbert, and to celebrate mathematics in the new millennium, the Clay Mathematics Institute of Cambridge, Massachusetts (CMI), named seven "prize problems", each one an important, classic question that has remained unsolved for some time. CMI has allocated a $1-million (34.1-million-baht) prize for each problem it posed.
One of the prize problems concerns proving the Riemann hypothesis, which many experts believe will puzzle us for centuries to come, despite the progress made by mathematicians to date.
And if it remains unsolved in 2009, it will have been an open question for 150 years, even though many great mathematical minds have worked towards its resolution. The problem is such a significant one that Hilbert himself said, "If I were to awaken after having slept for a thousand years, my first question would be: has the Riemann hypothesis been proven?'
The primes This article will not investigate the Riemann hypothesis in depth, as the mathematics involved quickly becomes too complicated for me to follow! It is concerned with the zeta function, which is connected very closely with the way that prime numbers are distributed.
Prime numbers are familiar to every schoolchild: they have exactly two natural number divisors, namely, 1 and the number itself. Therefore, 13 is a prime number because it has no factors other than 1 and 13; but 15 is not a prime number because it can be divided by 1, 3, 5 and 15.
One of the experts working on the zeta function is Marcus du Sautoy, a professor of mathematics at the University of Oxford. Professional mathematicians have quite a reputation for behavioural quirks, and in Prof du Sautoy's case, it is an obsession with prime numbers. His house number is prime, his football team all wear prime numbers on their shirts, and the telephone companies in England have become quite frustrated when he rejects any telephone number that isn't prime!
Primes in nature I was lucky enough to meet Prof du Sautoy two years ago when I travelled to Hong Kong with two teams of pupils from my school who were participating in the Southeast Asia Mathematics Competition. In his first lecture, he asked us to consider who first discovered the prime numbers. The surprising answer was that, long before the ancient Babylonian, Egyptian or Chinese mathematicians, prime numbers were being used to great advantage by a large, winged insect called a cicada.
Cicadas spend much of their life cycles buried under the earth and emerge periodically to mate. Although many have shorter lifespans, some species only emerge every 13 years, and others every 17 years. This adaptation to a prime-number time span helps cicadas avoid predators; a 15-year span would leave the insects vulnerable to predators with a three- or five-year cycle. The prime-number time span means that predators cannot easily prey upon the cicadas.
Patterns Professor du Sautoy described prime numbers as atoms, the building blocks of other numbers. For example, the number 66 is the product of 33 and 2. Two is prime, but 33 is not, as it has factors of three and 11. Breaking down 66 into its prime factors yields 2x3x11.
He is searching for a pattern to the prime numbers, or some sort of formula to predict where the next one will lie. When you look at individual prime numbers, they seem quite random.
Sometimes there are several close together, but then there will be a large gap before the next one. This lack of predictability is used to make online financial transactions more secure, an idea that I will explore in a future article.
Although the distribution of primes appears random, mathematicians have discovered that they follow some well-defined laws. The French philosopher and monk Marin Mersenne (1588-1648) found a formula for finding large prime numbers. At the moment, the largest known prime number is a Mersenne prime, the length of which would take six weeks to read out the digits!
A cash prize of $100,000 (3.41 million baht) is for a prime with 10 million digits.
For a complete listing and commentary on Hilbert's 23 problems, visit http://tinyurl.com/235oxz .
Activity: Join the search for the next large prime by visiting http://www.mersenne.org/ and downloading the free software. The prize might be smaller than the $1 million for proving the Riemann hypothesis, but the mathematics is considerably easier. Good luck!
Catherine Johnson is the head of mathematics at Shrewsbury International
School. For more information or
comments, you may email her at:
catherine.mathematics@gmail.com .
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