COMMENTARY
Playing dice with primes
Siddhartha Gadgil
Yitang Zhang, an unheralded mathematician in his 50s at the University of New
Hampshire, USA, who previously had to
work as an accountant and also at a
Subway restaurant, has stunned the
mathematical world by proving that
primes come in pairs. This was a weaker
form of a celebrated mathematical problem, the twin prime conjecture, which
asks whether there are infinitely many
numbers p so that both p and p + 2 are
primes. Zhang could not find twins,
merely distant cousins – he showed that
there is a fixed number N, at most about
70 million, so that there are infinitely
many pairs of primes p and q that differ
by at most N.
At almost the same time, Harald Helfgott, a young mathematician working in
Paris, showed that every odd number
greater than five is the sum of exactly
three primes (the so-called odd Goldbach
conjecture), a weaker form of another
celebrated problem, the Goldbach conjecture, which asks whether every even
number greater than two is the sum of
two primes. This improved a result from
an year ago of Terence Tao, who had
shown that odd numbers are sums of at
most five primes.
On seeing such results, one may ask:
(i) How can problems involving little
more than basic arithmetic be so hard?
(ii) Why should we care about such
results anyway?
The first question first. It is a remarkable result of Matijasevic that every
mathematical problem can be formulated
as an arithmetic problem, specifically
whether an equation with integer coefficients (e.g. 3x2 + 12y2 – 3214z12 +
32x = 0) has solutions with all variables
integers. For instance, we can write an
equation of this form, so that it has an
integral solution exactly if there is an algorithm to factorize integers in polynomial time (and another having solutions
exactly if all global solutions to the
Navier–Stokes equation are smooth).
This result is the climax of a series of
fundamental results in logic and computability, beginning with the work of
Church, Gödel and Turing.
This does not mean, of course, that
specific problems about numbers are
interesting, whether or not they are con162
sidered so by mathematicians – interesting problems from elsewhere will usually
have strange and complicated number
theoretic formulations. However, what it
does mean is that principles underlying
nature often manifest themselves in the
study of numbers, and the tools illuminating deep principles underlying numbers are likely to cast a wide light. So the
solutions to number theoretic problems
are of great value if they are consequences of a deep understanding of principles underlying the numbers, rather
than some accidental feature of the specific problem.
Indeed Zhang’s result is the consequence of such a principle, beautifully
explicated by Tao, who, along with Ben
Green, used it to prove another celebrated result in number theory – that
primes contain arbitrarily long arithmetic
progressions. Crudely, this principle
suggests that we should think of primes
like molecules in a gas.
Method from madness
A bottle full of gas has so many particles
that trying to understand it by solving the
governing equations will not get us very
far. Yet this very disorder leads to order.
While it is essentially impossible to
predict the trajectory of a particle, we
can predict that the time it spends in any
part of the container is proportional to
the volume of that part of the container –
the particle is equidistributed. Such a
view is the basis of statistical physics
(whose successes include explaining
previously empirical laws of thermodynamics), Einstein’s theory of Brownian
motion, and many other theories.
If we look at a molecule more closely,
taking into account not just where it is
but its velocity, which way it is pointing
and so on, we get a path not in the physical container but in phase space. A given
particle may not cover all of phase
space – for instance its energy may be
conserved, and perhaps its angular momentum as well. But remarkably, if we
allow for such conservation laws, then
the particle is (in an appropriate sense)
equidistributed in phase space. This can
be formulated precisely as ergodicity and
mixing properties of the dynamics of the
particle. Indeed, assuming a form of randomness, more precisely the so-called
Markov properties, lies at the heart of
Google’s Translate.
Structure and randomness in
mathematics
Primes clearly have order – for instance,
all primes greater than two are odd. This
order can be viewed in terms of mathematical structure. What one can postulate
(as explicated by Tao) is that, just as a
gas is random after taking into account
conserved quantities, primes are random
after taking into account appropriate structures. The randomness we seek is not that
of traffic on Indian roads – leading to
unpredictability, but that of a gas, with
mixing properties giving global order
from local disorder. Such mixing properties were earlier discovered in many other
mathematical contexts.
So what does this say about twin
primes? It is useful to view the twin
prime conjecture as an instance of a general question: for which sets S of natural
number are there infinitely many integers
n so that both n and n + 2 are contained
in S? This clearly depends on how big S
is, more precisely the density of S – the
fraction of elements between 1 and n that
are contained in S (for n large). In the
case of primes, the prime number theorem (from the 1800s) tells us that this
proportion is approximately 1/log(n).
For a set S with a lot of structure, such
as the set of squares, it is straightforward
to determine whether there are infinitely
many twin pairs. What if we take a random set, with the same density as primes?
Then the chance that a number k chosen
at random between 1 and n is prime is
about 1/log(n), so the chance that both k
and (k + 2) are prime is about (1/log(n))2.
But there are about n such pairs, so the
expected number of twin primes is
n/(log(n))2. As n becomes larger, this
goes to infinity, so there are infinitely
many twin primes. This means that a random set with the same density as primes
will satisfy the twin prime conjecture.
On the other hand, by the same argument such a random set will contain
CURRENT SCIENCE, VOL. 105, NO. 2, 25 JULY 2013
COMMENTARY
A little theory is a dangerous thing
inifinitely many pairs of adjacent numbers (n, n + 1). This is clearly not the
case for primes, as one of n and n + 1
must be even and so not a prime unless
n = 2. So we need a refinement, that
primes are random enough, after understanding and then taking account the underlying structure. It is indeed such
results that led Zhang, and earlier Tao
and Green, to their deep results.
more fruitful to seek the kind of randomness that leads to order.
So what are the lessons for science? I
would postulate the following: to the
extent that there is a genuinely useful
theory, such as Newton’s laws of motion,
we should of course seek it. But beyond
this, rather than building increasingly baroque models that are increasingly poor
approximations to the observed, it is
Siddhartha Gadgil is in the Department
of Mathematics, Indian Institute of Science, Bangalore 560 012, India.
e-mail: [email protected]
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