Notes from the history of Maths: Size matters...

On January 25th 2013, in Orlando, Florida, a computer running as part of the Great Internet Mersenne Prime Search (GIMPS) discovered the latest, largest known prime number.

They are called Mersenne primes after Marin Mersenne (1588-1648), a French monk. He acted as a communication hub for mathematicians and scientists of the day, sharing ideas between the likes of Descartes, Fermat, Pascal, Huygens and Galileo. Born to a working class family he went to the same school as Descartes. Eventually he joined “The Order of Minims”, who considered themselves the least (minimi) of all religions on earth and lived a very simple life.

One of his works was “L'harmonie universelle”, where he was the first to publish the laws relating to the vibrating string: its frequency being proportional to the square root of the tension, and inversely proportional to the length.

At the time, many mathematicians were obsessed with finding a pattern in prime numbers. Marcus du Sautoy, in “The Music of the Primes”, suggests that his interest in music may have given him insight into the formula for Mersenne primes: 2ⁿ-1. If you double the frequency of a note, you go up an octave, creating harmonic notes. A shift of 1 might be expected to create a very dissonant note, not compatible with any previous frequency – a ‘prime’ note. However, the formula could also have been a result of the search for perfect numbers. A perfect number, such as 28, is the sum of its factors other than itself (1,2,4,7,14).

It was Euclid that showed that whenever the sum of powers of 2 is a prime number, then you can create a perfect number by multiplying the sum by the highest double added. Since the sum of powers of two is 2ⁿ – 1 (sum of a Geometric Series), in modern notation, Euclid showed that:
Whenever 2ⁿ – 1 is prime, then (2ⁿ – 1) x 2ⁿ-1 is perfect.

For example, 1 + 2 + 4 = 7 is prime, so 7 x 4 = 28 is perfect. However, 1 + 2 + 4 + 8 = 15 is not prime, so no perfect number can be generated. The next sum works (31) and this generates a perfect number (31 x 16 = 496).

So, the hunt for perfect numbers - which were felt to have religious significance -  became a search for when 2ⁿ – 1 was prime. In 1644, Mersenne conjectured that this was the case when n = 2, 3, 5, 7, 13, 19, 31, 67, 127 and 257. It was quite a feat, at the time, to have found that 2047 (2¹¹-1) is not prime as 2047 = 23 x 89. No one knows how Mersenne came up with the list and it was only in 1876 that Edouard Lucas devised a method for checking Mersenne numbers.

He found 2⁶⁷-1 was not prime but 2⁶¹-1 was. Some have suggested this was a misprint in the original publication! It was also found that some numbers not on the list do produce primes (89 and 107). It was found that n=127 works and this remained the largest Mersenne prime until computers were invented. Only in 1952 was it found that 2⁵⁷ did not work. Whilst the Lucas method can show whether a Mersenne number is a prime or not, it doesn’t show how non-primes can be factorised. In 1903, Frank Cole gave a talk at the American Mathematical Society. Without saying a word he wrote:

2⁶⁷ – 1 = 193,707,721 x 761,838,257,287.

He got a standing ovation.

It is very difficult to factorise large numbers with large prime factors. This is why the hunt for ever bigger prime numbers is important. Prime numbers are used in on-line purchases. Credit card numbers are encoded by using numbers that can only be factorised into two primes of, at least, 60 digits each.

As e-commerce has grown, so has the need for bigger primes. The latest 2^57,885,161-1 has 17,425,170 digits.There is prize money available for large primes and the Electronic Frontier Foundation (www.eff.org) is now offering $150,000 for the first prime over 100 million digits and $250,000 for one over a billion digits. You can help the search by downloading the GIMPS software (www.mersenne.org). They will award you $3,000 for a new prime with less than 100 million digits. Happy prime hunting!

Don Hoyle

Mathematics Matters

The end is nigh: notes on the History of Maths

For most teachers this date is the end of term but could it be the end of the world as well?

If you search this date on the internet you’ll get nearly 2 billion results. There is even a web-site specifically dedicated to this date and the doom laden prophecies associated with it.

The main source for all this is the ancient Mayan calendar. The Mayan civilisation was dominant in Central America from about 200 BC to 1540 AD. They used 3 different calendars. There was a religious one, the Tzolk’in, based on a year of 260 days (20 lots of 13 days) and a secular one, the Haad, based on 365 days. This was made of 18 months, called ‘uinals’, of 20 days. To this was added 5 days called ‘the period with no name’ symbolised by chaos. You were thought to be cursed for life if you were born during these 5 days. The religious and secular calendars synchronised once every 52 years, about the length of a lifetime. This is an interesting exercise in calculating the Lowest Common Multiple of 260 and 365 (=18980 days or 52 secular years).

Image copyright of Roy Niswanger

The Maya were phenomenal at astronomical calculations even without the means to measure parts of the day. Their solar year works out to be equivalent to 365.242 days compared to our current value of 365.242198. It is more accurate than the Gregorian calendar we use (365.245 days). The Gregorian calendar, introduced by Pope Gregory XIII in 1572, attempted to allow for the solar year being just under 365 ¼ days long (as used in the Julian calendar from 45BC which brought in the leap year). The Gregorian calendar leapt forward 10 days and said leap days would not happen every century, unless the year was divisible by 400 (so 2000 was a leap year but not 1900). It took a while for some to adopt the Gregorian calendar, especially non-Catholic countries. It didn’t happen in Britain until 1752 when there were riots as people demanded to have their stolen days back.

Mayan name	B’ak’tun	K’atun	Tun	Winal	K’in
Days	144,000	7,200	360	20	1
Approximate solar years	394	20	1
Note that the system isn’t uniformly base 20 with a tun being 18 x 20 not 202

The Maya had another calendar, used for long periods. It was based on a year of 360 days, a tun. A k’atun was 20 tun and a b’ak’tun was 20 k’atuns or about 394 years. On 21^st December, we will simply move onto the next b’ak’tun.

The date 21^st December 2012, in the Mayan long calendar, will be 13.0.0.0.0.  Suggestions that there will be a cataclysm are generally denied by modern scholars of Mayan history, so you’ll need to prepare the lessons for next term!

Don Hoyle