11185272 whats next

Mersenne primesare prime numbersof the form

2 p  −  1

, where is necessarily a prime number (so these are prime Mersenne numbers). For example, 127is a Mersenne prime since 2 7  −  1 = 127. The largest known Mersenne prime tends to also be the largest known prime number. Currently, the largest known Mersenne prime is 2 77232917  −  1and has in excess of 24 million decimal digits.^[1]

Theorem.

For a number of the form

2 n  −  1

to be prime, it is a necessary condition that be prime. This is to say that if is composite, then so is

2 n  −  1

.

Proof. Consider the powers of 2 modulo a number of the form

2 n  −  1

: we have

1, 2, 4, 8, ..., 2 n   −  2, 2 n   −  1, 1, 2, 4, 8, ...

and so on and so forth, showing that an instance of 1 is encountered periodically at every doubling steps. This means that for any positive integer , the congruence

2Ȁ Tag Archives: mersenne At the end of this post, I made a totally naive guess that the recently discovered candidate to be the , the 45th Mersenne prime, would have 10.5 million digits. There was absolutely no systematic basis for that guess, but I did suggest having an office pool for the number of digits, so what I lack in mathematical sophistication is made up for by my instinct for good nerd party games. On the other hand, Isabel at God Plays Dice predicted 14.5 million digits based on a number theoretic argument. Since I am merely a wannabe number theorist, I can’t compete with that sort of thing. But I can make up a mean Excel spreadsheet, so I figured I’d do a little data plotting and see what happened. If you make a plot of the number of digits in , the nth Mersenne prime, going all the way back to antiquity, here’s what you get: The horizontal axis is n and the vertical axis is the number of digits in . Admit it — one look at this plot and you’re itching to add some trendlines. Here’s what you get when you add both an exponential trendline (perhaps the obvious choice given the shape) and a 6th-degree polynomial: Th News from the world of maths: Today is brought to you by the number 11,185,272... ...and yesterday was brought to you by the number 54, thanks to the Mathematical Association of America's NumberADay blog. Every working day they post a number and a biography of its interesting properties. Today's number 11,185,272 is the number of decimal digits in the 46th known Mersenne prime, discovered on Sept. 6, 2008 (you can read more in Prime record broken? on Plus). 54 might seems less significant, but in fact thanks to the MAA Plus now knows that it is the smallest number that can be written as the sum of 3 squares in 3 ways, the number of colored squares on a Rubik’s cube, and is a nonadecagonal (19-gonal) number! posted by Plus @ 2:12 PM Mersenne Primes: History, Theorems and Lists Contents: Early History Perfect Numbers and a Few Theorems Table of Known Mersenne Primes The Lucas-Lehmer Test and Recent History Conjectures and Unsolved Problems See also Where is the next larger Mersenne prime? and Mersenne heuristics 1. Early History Many early writers felt that the numbers of the form 2n-1 were prime for all primes n, but in 1536 Hudalricus Regius showed that 211-1 = 2047 was not prime (it is 23.89).  By 1603 Pietro Cataldi had correctly verified that 217-1 and 219-1 were both prime, but then incorrectly stated 2n-1 was also prime for 23, 29, 31 and 37.  In 1640 Fermat showed Cataldi was wrong about 23 and 37; then Euler in 1738 showed Cataldi was also wrong about 29.  Sometime later Euler showed Cataldi's assertion about 31 was correct. Enter French monk Marin Mersenne (1588-1648).  Mersenne stated in the preface to his Cogitata Physica-Mathematica (1644) that the numbers 2n-1 were prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257  and were composite for all other positive integers n < 257.  Mersenne's (incorrect) conjectu