Terence Tao is a Professor at the Department of Mathematics, UCLA
and one of Australia's most acclaimed mathematicians. Indeed, he is
arguably the world's greatest living mathematician. In 2006, he was
awarded a Fields Medal, which is the top prize a mathematician can win, and at 24 became the youngest ever full professor at UCLA.
I recently went to Tao's Clay–Mahler Lecturer at UNSW, which was a fascinating look at prime numbers. I managed to grab Terence for a quick chat.
Primes are integers that can only be divided by themselves and one.
For example, the number 10 can be divided by 1, 2, 5 and 10 - whilst
the number 11 can only be divided 1 and 11. The first few primes are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 ....
One
of the interesting things about the primes is that there is no known
formula yielding all of them - you can't simply plug a few numbers into
a formula to generate a list of the primes. However, on a large scale,
their distribution can be modelled. The primes behave as if they are
distributed pseudorandomly - see the picture on the right. Each dot in
this Ulam spiral
represents a prime number - you start in the middle, and wind outwards
like a spiral - each dot is a prime, whilst empty space is a non-prime.
Whilst you can see various patterns, nothing is predictable.
The prime number theorem says that the probability of a given number n being prime is inversely proportional to its logarithm. Euclid proved that there are infinitely many prime numbers way back in 300BC - see Euclid's Theorem for more. The current largest known prime was discovered in 2008 by the distributed computing project Great Internet Mersenne Prime Search and has 12,978,189 digits:
243,112,609 − 1.
Primes are very important for public-key cryptography
- that is, the way your credit card numbers are encrypted in online
transactions. The cryptography makes use of the fact that is difficult
to factorise large numbers
into their prime factors, whilst it is comparatively easy to multiply
two large primes together. No efficient integer factorisation algorithm
is currently known - in 2005 a 193-digit number was factorised, but it
took 5 months.
Terence Tao, along with Ben Green, proved that the sequence of prime numbers contains arbitrarily long arithmetic progressions - this is the Green-Tao theorem. What this means is that for any number k, there is an arithmetic progression of primes k
long. An arithmetic progression is one in which the difference between
two numbers in the progression is the same. For example, the series 2,
4, 6, 8, 10... is an arithmetic progression with common difference 2.
Green and Tao proved that such sequences exist within the primes for
any length of series you want. For example, the series 3, 7, 11 is a
prime sequence of length 3 with common difference 4. The series 3, 5, 7
is length 3 with common difference 2. The current record is a series of
25 primes.
I have just finished reading the excellent book The Music of the Primes by British author Marcus du Sautoy - I highly recommend it. It details the story of the Riemann hypothesis
which is considered by many to be the most important unresolved problem
in mathematics. A solution to the Riemann hypothesis could make an
immense contribution to our understanding of the distribution of prime
numbers. You certainly don't need to be a maths geek to understand this
book - it is a great historical tale. You can buy the book from Amazon
by clicking on the cover on below.
God may not play dice with the universe, but something strange is going on with the prime numbers - Paul Erdos
