Laura's fingernails show a prime number in binary form. What is the next higher prime? Further down on this page, you'll get a chance to submit your answer. The diagram on the right shows part of the CRT display on a "high speed" computer of the 1950's (the Ferranti Mark 1). The last line represents the number 262,673 in binary form (it is not a prime, but =193 x 1361).

In those days, computers had no fancy input-output devices, so programmers used to read code and data directly in binary. To make that less confusing, the Mark 1 showed the "bits" in groups of 5, each group denoting one of 32 "digits" , say, 0,1, ... , 9, A,B, ... ,V. In that language, Laura's left hand would signal "9". Today, those who deal with the guts of computers use groupings of 4 bits and the 16 digits 0, ... , 9, A, ... ,F. This notation is called "hexadecimal" (meaning base 16), and these digits are shown in pairs known as "bytes". The highest number expressible in one byte is 255, written as FF. Binary numbers are all over today's digital technology, because switches of any kind are most apt to have two states: ON or OFF.

While binary or hexadecimal refer to the form of a number, primeness (or "primality") relates to its substance. Prime numbers are increasingly important in applications such as cryptography -- the science of making messages unreadable to the enemy -- of obvious interest to both corporations and the military. Their offspring finite fields are basic to coding theory -- the science of restoring garbled messages. Much of today's world, from ordinary banking machines to complex missile systems, depends on such techniques.

Hunting for primes is not easy: they seem to be scattered among integers in a more or less random pattern, although there are some striking over-all regularities. If you sort primes by their last (decimal) digit, you'll find that, in the long run, each of the 4 possible digits 1, 3, 7, 9 garners exactly 25% of the primes -- and the same phenomenon occurs when you use a base other than 10. This is a famous theorem of Dirichlet, which at least partly responsible for the pattern shown on the right. Moreover, if you randomly pick a number with n digits, the probability that you got a prime is approximately 0.43 divided by n. This the famous Prime Number Theorem (the 0.43 is actually log e = 1/ln 10).

In contrast to these global regularities, the local distribution of primes is completely erratic. Between successive primes there are gaps of abitrary length. On the other hand, the smallest possible gap (length 1, as in 41,43) seems to occur infinitely often -- but no one has yet proved that this is actually the case. Surprisingly, there are formulas for grinding out primes, but they are useless in practice. In other words, there is still a lot to do (and even money to be made) in the field of good prime hunting.

The Pacific Institute for the Mathematical Sciences (PIMS) is a non-profit organisation supported by five universities of Western Canada and dedicated to the promotion of mathematical research -- but it also has a programme of education and public awareness.

This contest is now over. The correct answer was 239 or 11101111. The winner of the draw was Chad Simpson.