Think of a number with 92 digits, then keep adding 210 until you get into the record books. Keith Devlin reports
The number of the beast

An Austrian mathematician, Manfred Toplic, has computed his way into the record books by finding a sequence of nine prime numbers that are all equally spaced, like the rungs on a ladder.

A prime number is any whole number that is only evenly divisible by itself and 1, such as 2, 3, 5, 7, 11, and 13 (the only primes less than 15). It is known that there are an infinite number of such numbers, which is enough of an excuse for the computationally minded to see who can identify the largest one. The record so far is a prime number of almost a million digits, discovered by a 19-year-old Californian student earlier this year (From the prime to the meticulous, Online, February 12). But for some prime number aficionados like Toplic, finding individual world record primes is too easy. They look for what mathematicians call an “arithmetic progression” of primes — a sequence of numbers that starts with some initial number p and increases by an equal step d, giving the numbers: p, p + d, p + 2d, p + 3d, and so on.

The prime numbers 3, 5, 7 give an arithmetic progression of primes of length 3. Over the years, armed with ever more powerful computers, mathematicians have found arithmetic progressions of primes of all lengths up to 8, that level being reached as recently as last November. The difference (d) between the primes in that progression was 210. The initial prime (p) was a giant with 92 digits.

Starting with that discovery, Harvey Dubner in America, Tony Forbes in the UK, and Nik Lygeros, Michel Mizony and Paul Zimmermann in France used the Internet to solicit computational help from around the globe in a search for nine consecutive primes in arithmetic progression. The idea was to divide up the search into different ranges of trial values.

On January 15, Toplic, one of the hundred or so volunteers who signed up, hit the jackpot. As with the previous record of eight primes in arithmetic progression, Toplic’s new sequence starts with a 92 digit prime and goes up in steps of 210. In this case, the initial prime number is: 99,679,432,066,701,086,484, 490,653,695,853,561,638,982,364, 080, 991,618,395,774,048,585,529, 071,475,461,114,799,677,694,651.

The team is now trying to find 10 consecutive primes in arithmetic progression. Dubner estimates that it could take 500 people some six months of heavy duty computing to find an answer. The purpose of this hunt? Pure entertainment. Think of it as similar to going for a world record in athletics.

19 February 1998