Alex's Adventures in Numberland - Alex Bellos [113]
The current record for largest prime is held by the 45th Mersenne prime: 243112609 – 1, which is a number almost 13 million digits long, found in 2008 by a computer connected to GIMPS at the University of California, Los Angeles. The 46th and 47th Mersennes found were actually smaller than the 45th. This happened because computers are working at different speeds on different sections of the number line at the same time, so it is possible that primes in higher sections will be discovered before primes in lower ones.
GIMPS’s message of mass voluntary cooperation for scientific advancement has made it an icon of the liberal web. Woltman has unintentionally turned the search for primes into a quasi-political pursuit. As a mark of the symbolic importance of the project, the Electronic Frontier Foundation, a digital-rights campaign group, has since 1999 offered money for each prime whose digits reach the next order of magnitude. The 45th Mersenne prime was the first to hit ten million digits and the prize money won was $100,000. The EFF is offering $150,000 for the first prime with 100 million digits, and $250,000 for the first prime with a billion digits. If you plot the largest-known primes discovered since 1952 on a graph with a logarithmic scale against the time of discovery, they fall on what is almost a straight line. As well as showing how the growth of processing power has advanced remarkably consistently over time, the line also allows us to estimate when the first billion-digit prime will be discovered. I’d put money on it being found by 2025. Writing out this number in type where each digit is a millimetre would stretch further than from Paris to Los Angeles.
Digits in highest-known prime by year of discovery.
With an infinite number of primes (whether there is an infinite number of Mersenne primes, however, is not yet known), the search for higher and higher primes is a never-ending task. Whatever prime number we reach, no matter how large, there will always be a prime number even larger taunting us for our lack of ambition.
Endlessness is probably the most profound and challenging idea of basic maths. The mind finds it difficult to cope with the idea of something going on for ever. What, for example, would happen if we start counting 1, 2, 3, 4, 5…and never stop? I remember asking this seemingly simple question as a child, and receiving no straightforward answer. The default response from parents and schoolteachers was that we get to ‘infinity’ but this answer essentially just restates the question. Infinity is simply defined as being the number that we get to when we start counting and never stop.
Nevertheless, we are told from a relatively early age to treat infinity like a number, a weird number, but a number all the same. We are shown the symbol for infinity, the endless loop 8 (called a ‘lemniscate’), and taught its peculiar arithmetic. Add any finite number to infinity and we get infinity. Subtract any finite number from infinity and we get infinity. Multiply or divide infinity by a finite number, as long as it isn’t zero, and the result is also infinity. The ease with which we are told that infinity is a number disguises more than 2000 years of struggling to come to terms with its mysteries.
The first person to showcase the trouble with infinity was the