## Exploring the number jungle online

 Heather Catchpole ABC Science Online Tuesday, 5 October  2004

 The number of sunflower seeds per row increases according to the Fibonacci sequence (Image: U.S. National Genetics Resources Program)
Everyone from mathematicians to astronomers to mobile phone companies are making use of an online database to handle the jungle of numbers they are confronted by in their daily work.

Dr Neil Sloane of AT&T Shannon Labs in New Jersey told the recent Annual Meeting of the Australian Mathematical Society in Melbourne that his On-Line Encyclopedia of Integer Sequences will soon contain 100,000 mathematical sequences and he is planning an e-party to celebrate it.

A mathematical sequence is an infinitely long list of numbers that can generally be calculated using simple rules.

The simplest sequences are used to calculate mortgage repayments or compound interest but probably the most famous sequence is the Fibonacci's sequence, which is formed by adding the last two numbers in the sequence. Its first few terms are 0,1,1,2,3,5,8,13,...

Sloane has been working on his encyclopaedia for 40 years. It started as a book to help mathematicians understand if a pattern of numbers was an actual sequence, or just a mess of random numbers.

Today, astronomers look for sequences, as signs of extraterrestrial intelligence, in the random patterns of radio signals received from space.

Sequences can also help mobile phone companies to make sure calls don't interfere with each other.

"When you make a call you have a frequency assigned that has to be assigned in such a way as to avoid collisions," he said.

Frequencies based on sequences of numbers that never repeat can ensure that the frequencies allocated to mobiles never repeat.

Sequences are also used to decode messages in cryptography.

Large primes and unsolved mysteries

Sloane said his database also lists the sequences of primes used to calculate the largest prime numbers, numbers that can only be divided by one and itself.

Although the number of primes is infinite, big ones are rare, he said. Sequence 43 in the encyclopaedia is a list of the Mersenne primes, primes that can be calculated by the formula '2p-1', where 'p' is a prime number.

The 41st known Mersenne Prime and the biggest known prime number, 224,036,583-1, was calculated by Josh Findley in May this year, using a screensaver that searches for large prime numbers.

Associate Professor James Franklin, from the University of New South Wales in Sydney said there were many unsolved sequences where there was a pattern, but no formula had been found to calcualte it.

"If you just take the primes they are distributed in a way that looks random, but there is an underlying pattern," he said. "But a lot of patterns initially just look like a mess."

Sloane's favourite messy sequence is Recamen's sequence, which begins the same as Fibonacci's sequence but is subtly different.

Recamen's sequence is difficult to analyse because it has no particular regularity, and doesn't increase, decrease or oscillate in any regular way.

"If you plot the Fibonacci numbers they grow at a predictable, smooth rate," he said. "Recamen's sequence is particularly mysterious. It oscillates like crazy, drifting up and dragging down. If you plot it, it looks more like a child's scribble."