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VOOZH | about |
Mersenne Primes are a subset of Mersenne numbers. Mersenne numbers are those which are generated by the formula Mn = 2n-1. These are named after Marin Mersenne who studied them in 17th century.
First 10 Mersenne numbers are:
M1 = 21-1 = 1 | M2 = 22-1 = 3 | M3 = 23-1 = 7 | M4 = 24-1 = 15 | M5 = 25-1 = 31 | M6 = 26-1 = 63 | M7 = 27-1 = 127 | M8 = 28-1 = 255 | M9 = 29-1 = 511 | M10 = 210-1 = 1023 |
A Mersenne prime is a prime number that can be expressed in the form Mn = 2n-1. To be considered a Mersenne prime, n itself must be a prime number. If n is composite, then Mn is sure to be composite, but if n is prime, then Mn may be prime or composite. (This test is known as Fermat primality test.)
For example: When:
For n = 3, M3 = 7, which is prime but for n = 11 (prime). Bu for n =11, M11 = 2047, which is not a prime.
First few Mersenne Primes are:
3, 7, 31, 127, 8191, 131071, 524287, 214748364.
First few values of n, for which 2n - 1 gives primes, are:
2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, . . .
The largest known prime number is also a Mersenne Prime: (as of October 2024)
Largest known prime: 282, 589, 933 ā 1, having 24, 862, 048 digits.
Largest 5 Mersenne prime are:
Rank | Number | Digits |
|---|---|---|
1 | 282589933 ā 1 | 24, 862, 048 |
2 | 277232917 ā 1 | 23, 249, 425 |
3 | 274207281 ā 1 | 22, 338, 618 |
4 | 257885161 ā 1 | 17, 425, 170 |
5 | 243112609 ā 1 | 12, 978, 189 |
Mersenne primes are searched using algorithm of GIMPS (Great Internet Mersenne Prime Search). This is formed in 1996 to discover new world record size Mersenne primes.
A perfect number is a positive integer that is equal to the sum of its proper divisors(except itself) for instance 6, is a perfect number because its divisors 1, 2, 3 add up to 6.
In the 4th century BC, Euclid demonstrated that if (2p - 1) is a prime number, then 2pā1(2p ā 1) is a perfect number. Later, in the 18th century, Leonhard Euler proved the reverse: all even perfect numbers follow this pattern. This result is known as the Euclid-Euler theorem.
In conclusion, Mersenne primes are a special type of prime number with a unique form, 2p ā 1, where p is also prime. These primes have fascinated mathematicians for centuries due to their connection to perfect numbers and their role in modern cryptography.
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