Mersenne primes are a special category of primes expressed as 2 to the "p" power minus 1, in which "p" also is a prime number. The theorems around mersenne primes turn out to be so aesthetic, that they had to have "come straight from the book". :)
Mersenne primes have a very interesting history:
[a] pre 1532
Mathematicians conjectured that all (2^n - 1) were primes for every prime n.
[b] 1532
Regius proved that (2^11 - 1) was not a prime.
[c] 1600
Cataldi proved (2^17 - 1) and (2^19 - 1) were both primes and conjectured that the theorem was true for primes 23, 29, 31, and 37.
[d] 1640
Fermat proved Cataldi was wrong about 23 and 37.
[e] 1644
Mersenne conjectured the theorem was true for primes 2, 3, 5, 7, 13, 17, 19, 31, 61, 127 and 257 and false for any other prime less than 258.
[f] 1947
It took excess of 3 centuries for folks to come to a mathematical conclusion about Mersenne's conjecture. Turned out, his conjecture was pretty close. He was right about all this primes, and had missed out 89 and 107. Took a lot of mathematicians, from Euler to our very own Ramanujam to verify Mersenne's theorem.
Subsequently, various folks have come up with tests for checking primality based on Mersenne primes (including the non-exponential primality test from IIT Kanpur) and with all the computing power available for brute force testing, what was discovered recently was the 43rd Mersenne prime.
Back in 1965 or so, math dept of Urbana cracked the Mersenne prime for n=11213. They were so kicked, they made a stamp out of it and would imprint it on all postal letters going out of the Urbana math dept (see below). This of course lasted until 1976 when Urbana math dept cracked the four-color theorem and proved it correct. After that for a while, the four-color theorem was on the envelopes.
More details about Mersenne Primes here
fun stuff.
Vinod
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