The stuff that Mersenne discovered about how the world ticks is written in many text-books and taken for granted now - merging into the invisible background of what everybody knows to be true. Although 99% of us couldn't even carry out those experiments reliably, let alone think them through. Père Mersenne is mostly remembered for the a particular class of prime numbers which he worked on but which were recognised by people before him and worked on obsessively and extensively by many people after him.
Take the formula n = 2p - 1 [or write it n = 2^p - 1].
You can prove that if n is prime; then p must be prime.
But you cannot prove that if p is prime; then n must be prime but in many cases it is true. Chekkitout:
The first prime numbers are 2 3 5 7 11 13 . . .
2^2-1 = 3 !prime!
2^3-1 = 5 !prime!
2^5-1 = 31 !prime!
2^7-1 = 127 !prime!
at which point you might think "we're on to something here" only to find that
2^11-1 = 2047 = 23 x 89 so not prime . . . bummer. But
2^13-1, 2^17-1 and 2^19-1 are prime and so it goes on. Mersenne claimed that he had checked all the prime p's for n = 2^p - 1, up to and including p=257. He said that p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257 were prime but that the other 44 primes in the range yielded a composite [ie not-prime] n.
In the process, Mersenne committed both type-I and type-II errors: he called some cases primes [61 and 257] when they weren't and also believed that some cases were composite [61, 89, 107] when later careful work showed they were prime. In fact there are many more examples of p where the formula results in composite numbers than there are Mersenne Primes. When I was born only 17 of them were known and M17 was much bigger than you could reliably write down on a bit of paper [686 digits]. Indeed there are even now only 48 known Mersenne Primes and the next one to be discovered will be astronomically large. In the formula 'p' is the exponent and exponential increase means that the tenth Mersenne Prime aka M10
2^89 - 1 = 618970019642690137449562111while the largest proven [hats off to Curtis Cooper in Feb 2013] current example is M48 = 2^57,885,161 - 1 which yields a number with more than 17 million digits.
If you have access to a supercomputer and 6 months of processing time, you can have a punt at finding the next Mersenne Prime M49. M37 was knocked off by Roland Clarkson in 1998 while he was a 19 y.o student in California (and not Berkeley or Stanford either). If you don't have a supercomputer you can join GIMPS (Great Internet Mersenne Prime Search) and have a go at finding M50. Disambiguation: GPS will find Dublin's orbital ring-road the M50. It's like a turkey shoot for geeks.
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