Prime numbers never cease to enthrall and fill us with wonder. There is something about them, I don’t know what, but if you catch the bug you will never stop being amazed. (Prime numbers are those that have no factors other than 1 or themselves, e. g. 7 or 13)
Take Mersenne numbers. They are named after the 17th century French monk who studied them and are of the form: 2^n – 1, or 2 raised to the power n for some positive integer n, minus 1. The first few are: 1, 3, 7, 15, 31, 63, 127, 255 …. Since 4 out of the above 7 (not counting the number 1, which is outside the realm of numbers eligible to be prime) are prime, Mersenne thought that this series would be a rich source of prime numbers. He conjectured that an infinite number of these numbers would be prime – something that may be true but hasn’t been proven yet.
But the Mersenne series does yield some interesting properties. First, the exponent of 2 (i. e. the power to which 2 must be raised in a Mersenne number) must itself be prime! And even then very few prime exponents actually result in a Mersenne prime. Even so, last year, the largest prime currently known to man. a Mersenne prime, was found using fast computer searching. It is the 48th Mersenne prime and has more than 17 million digits!
The largest prime known: 257,885,161 − 1, a Mersenne prime.
(If you can be the first to find a prime number with more than a hundred million digits I believe there is a prize of $150,000 awaiting you – so go at it!)
Mersenne numbers are also an example of an area of mathematics called Dynamical Sequences. Here you generate numbers, starting with a seed, such as zero and repeatedly plugging into a simple algebraic expression. For example you can generate the Mersenne numbers by using the expression 2x+1. Start by plugging x=0 and the expression calculates out 1. Now plug 1 into the expression and you get 3. Repeat and you get 7 followed by 15, 31, 63, 127 … i. e. the Mersenne numbers.
Holly Krieger, currently at MSRI* (see below) in Berkeley, is studying the arithmetic properties of some magical dynamical sequences and coming up with mind blowing results! MSRI has a video series at the website Numberphile.com full of fascinating math games, puzzles, tricks and number facts. To see a wonderful show on her dynamical sequences and how prime numbers crop up everywhere see her presentation by clicking her picture below. Do see it all the way to the end for she has an unexpected surprise!
*MSRI – Mathematical Sciences Research Institute, Berkeley. Website here. MSRI is an advanced institute located in the Berkeley hills dedicated to the advancement of fundamental math and also to the cultivation of its beauty, power and importance.
I have the privilege to support some of their activities and in return get a rich exposure to fun math people and cutting edge researchers in the field. Next time you’re in Berkeley check them out. Ashok