Enter an exponent n to test if 2^n - 1 is a Mersenne prime
Exponent n
Result
Mersenne Number
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Verdict
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Detailed Derivation
Mersenne Prime Properties
Mersenne number: M(n) = 2^n - 1
If M(n) is prime, it is a Mersenne prime
n must be prime for M(n) to possibly be prime
Lucas-Lehmer test: S(n-1) mod M(n) = 0
Mersenne primes are prime numbers of the form 2^n-1. They are the largest known primes and are intimately connected to perfect numbers through the Euclid-Euler theorem. The GIMPS project continues to discover larger Mersenne primes.
⚠For n greater than 20, the Mersenne number becomes very large. The Lucas-Lehmer test is efficient even for very large Mersenne numbers.
What Is a Mersenne Prime?
A Mersenne prime is a prime number that is one less than a power of two. They are named after Marin Mersenne, a 17th-century French monk and mathematician. Mersenne primes are the largest known primes and have a special connection to perfect numbers.
Primality Condition
If 2^n-1 is prime, then n must be prime. But the reverse is not always true: 2^11-1=2047=23x89 (n=11 is prime, but M(11) is composite).
Lucas-Lehmer Test
A special primality test for Mersenne numbers. Much faster than general primality tests for numbers of this form. Used by GIMPS.
Perfect Number Link
Each Mersenne prime generates an even perfect number: P = 2^(n-1) x (2^n-1). This connection was known to Euclid and proved by Euler.
GIMPS
Great Internet Mersenne Prime Search is a distributed computing project searching for Mersenne primes since 1996. Volunteers worldwide contribute computing power.
Teaching Example: n=7. M(7)=2^7-1=128-1=127. Lucas-Lehmer: S(1)=4. S(2)=4^2-2=14. S(3)=14^2-2=194 mod 127=67. S(4)=67^2-2=4487 mod 127=42. S(5)=42^2-2=1762 mod 127=111. S(6)=111^2-2=12319 mod 127=0. Zero reached at S(n-1)=S(6). 127 is Mersenne prime! Perfect number: 2^6x127=8128.
Applications
Prime ResearchPerfect NumbersCryptographyDistributed ComputingNumber TheoryRandom Generation
FAQs about Mersenne Primes
What is a Mersenne prime?▼
A prime of the form 2^n-1. First four: 3,7,31,127. 51 known, largest has 24 million digits (2^82589933-1).
What is the Lucas-Lehmer test?▼
S(1)=4, S(k+1)=S(k)^2-2 mod M. If S(n-1)=0 mod M, M is prime. This test is very efficient for Mersenne numbers.
Are all 2^n-1 prime when n is prime?▼
No. Counterexample: n=11 is prime but 2^11-1=2047=23x89 is composite. n must be prime for 2^n-1 to possibly be prime, but it is not sufficient.
How are Mersenne primes related to perfect numbers?▼
Euclid-Euler: If M=2^n-1 is prime, then 2^(n-1)xM is a perfect number. 2^2-1=3 gives 6. 2^3-1=7 gives 28. All even perfect numbers have this form.
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