| Mersenne Prime |
Website Links For Mersenne |
Information AboutMersenne Prime |
| CATEGORIES ABOUT MERSENNE PRIME | |
| prime numbers | |
| articles containing proofs | |
M The central question about these numbers is which of them are also at Martin.. Mersenne primes were considered already by Euclid, who found a connection with the Perfect Number s. They are named after 17th century French scholar Marin Mersenne , who compiled a list of Mersenne primes with exponents up to 257. His list was only partially correct, as Mersenne mistakenly included ''M''67 and ''M''257 (which are composite), and omitted ''M''61, ''M''89, and ''M''107 (which are prime). Mersenne gave no indication how he came up with his list, and its rigorous verification was completed more than two centuries later. Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether there is a largest Mersenne prime, which would mean that the set of Mersenne primes is finite. The Lenstra-Pomerance-Wagstaff Conjecture asserts that, on the contrary, there are infinitely many Mersenne primes and predicts their Order Of Growth . Perhaps even more embarrassingly, it is not known whether infinitely many Mersenne numbers with prime exponents are Composite , although this would follow from widely believed conjectures about prime numbers, for example, the infinitude of Sophie Germain Prime s. A basic theorem about Mersenne numbers states that in order for ''M''''n'' to be a Mersenne prime, the exponent ''n'' itself must be a prime number. This rules out primality for numbers such as ''M''4 = 24 -1 = 15: since the exponent 4=2×2 is Composite , the theorem says that 15 is also composite; indeed, 15 = 3×5. The three smallest Mersenne primes are : ''M''2 = 3, ''M''3 = 7, ''M''5 = 31. Whilst it is true that only Mersenne numbers ''M''''p'', where ''p'' = 2, 3, 5, … ''could'' be prime, it may nevertheless turn out that ''M''''p'' is not prime even for a prime exponent ''p''. The smallest counterexample is the Mersenne number : ''M''11 = 211 − 1 = 2047 = 23 × 89, which is not a Mersenne prime, even though 11 is a prime number. The lack of an obvious rule to determine whether a given Mersenne number is prime makes the search for Mersenne primes an interesting task, which becomes difficult very soon, since Mersenne numbers grow very fast. The Lucas–Lehmer Test For Mersenne Numbers is an efficient Primality Test that greatly aids this task. Search for the largest known prime has somewhat of a cult following. Consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using Distributed Computing . SEARCHING FOR MERSENNE PRIMES The identity : | ||
| { Class | "wikitable" |
|
|