# Take one blackboard, one piece of chalk, 1,257,787 twos and work out a world record (don't forget to subtract one)

Yesterday was a bad day for (2859,433-1). Listed in the Guinness Book of Records as the largest-known prime number, it has been knocked into numerical obscurity by (21,257,787-1), a monster of 378,632 digits which has been proven prime by a computer at Cray Research, Wisconsin. It would take about 12 pages of this newspaper to print out the number in full. The old record-holder would have run out after about nine pages.

Prime numbers - those that can be divided without remainder by no whole numbers other than one and themselves - have fascinated mathematicians for more than 2,000 years. Euclid provided the first simple proof that there is an infinity of primes. (If not, just multiply them all together and add one. The resulting number is either itself prime or has a prime divisor different from those you started with. QED.)

For the last 100 years, we have even known roughly how many prime numbers there are below any given figure. (This is given by the so-called prime number theorem, first proved in 1896.) Yet despite knowing there is no largest prime, people have continued searching for ever larger ones.

In 1772, the record was held by a 10-digit number; by 1884, it had been raised to 20 digits, but the real acceleration began in the computer age. In 1971, months of computer calculations led to the discovery of a 6,002- digit prime, and in the 1980s and 90s Cray computers have been pushing the record higher almost every year.

A spokesman for Cray described prime testing as a "torture test" for supercomputers. Others might call it a waste of time. Recently, however, the task of factorising large numbers has had important applications in computer security.

We had intended to print the new top prime in full, but - perhaps for reasons of security - the people at Cray have not divulged all its digits. Of course, you can work it out yourself: just take 1,257,787 twos, multiply them together and subtract one from the answer.

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