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 |
Definition of Mersenne Prime
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, . . .
Largest Mersenne Prime
The largest known prime number is also a Mersenne Prime: (as of October 2024)
Largest known prime: 282, 589, 933 − 1, having 24, 862, 048 digits.
Largest 5 Mersenne prime are:
Rank | Number | Digits |
|---|---|---|
1 | 282589933 − 1 | 24, 862, 048 |
2 | 277232917 − 1 | 23, 249, 425 |
3 | 274207281 − 1 | 22, 338, 618 |
4 | 257885161 − 1 | 17, 425, 170 |
5 | 243112609 − 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.
Relationship of Mersenne Primes with Perfect Numbers
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 2p−1(2p − 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.
Conclusion
In conclusion, Mersenne primes are a special type of prime number with a unique form, 2p − 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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