Mersenne prime facts for kids

A Mersenne prime is a special kind of prime number that is one less than a power of two. Imagine taking the number 2, multiplying it by itself many times, and then subtracting 1. If the result is a prime number (a number only divisible by 1 and itself), then it's a Mersenne prime! These numbers are named after Marin Mersenne, a French monk who studied them in the 1600s.

Mersenne primes are very important in mathematics. They have a close connection to perfect numbers, which are numbers equal to the sum of their own divisors (like 6 = 1 + 2 + 3). Many of the largest known prime numbers are Mersenne primes because there are special tests that make them easier to check for primality.

Currently, 52 Mersenne primes are known. The largest known prime number is a Mersenne prime: 2^136,279,841 − 1. Since 1997, many new Mersenne primes have been found by a huge online project called the Great Internet Mersenne Prime Search (GIMPS), where people around the world use their computers to help search.

What Are Mersenne Primes?

A Mersenne number is a number that can be written as M_n = 2ⁿ − 1, where n is a whole number. For example:

• If n = 2, then .
• If n = 3, then .
• If n = 5, then .

A Mersenne prime is simply a Mersenne number that is also a prime number. So, 3, 7, and 31 are Mersenne primes.

For a Mersenne number to be prime, the exponent n must also be a prime number. If n is a composite number (meaning it has factors other than 1 and itself), then M_n will also be composite. For example, if n = 4 (which is composite, 4 = 2 × 2), then , and 15 is composite (3 × 5).

However, just because n is a prime number doesn't automatically mean that M_n will be prime. The smallest example of this is when n = 11. M₁₁ = 2¹¹ − 1 = 2047. If you try to divide 2047, you'll find that 2047 = 23 × 89. So, 2047 is not a prime number.

The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, and so on. (sequence A000043 in OEIS) The actual Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, and so on. (sequence A000668 in OEIS)

Mathematicians are still trying to figure out many things about Mersenne primes. For example, it's a big mystery whether there are infinitely many Mersenne primes or if the list eventually stops.

Mersenne Primes and Perfect Numbers

Mersenne primes are very closely linked to perfect numbers. A perfect number is a positive whole number that is equal to the sum of its positive divisors, not including the number itself. For example, 6 is a perfect number because its divisors (other than 6) are 1, 2, and 3, and 1 + 2 + 3 = 6.

In ancient Greece, the mathematician Euclid proved a special rule. He showed that if 2^p − 1 is a Mersenne prime, then 2^{p − 1}(2^p − 1) will always be a perfect number.

Much later, in the 1700s, Leonhard Euler proved the opposite. He showed that every even perfect number must be in this exact form. This amazing connection is known as the Euclid–Euler theorem. We still don't know if there are any odd perfect numbers!

The Story of Mersenne Primes

Mersenne primes are named after Marin Mersenne, a French scholar from the 1600s. He made a list of what he thought were Mersenne primes with exponents up to 257. His list from 1644 included:

2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257.

Mersenne's list was mostly correct for the smaller numbers. However, he made some mistakes. He incorrectly included M₆₇ and M₂₅₇ (which are not prime). He also missed M₆₁, M₈₉, and M₁₀₇ (which are prime). Mersenne didn't explain how he created his list.

It took many years for mathematicians to check Mersenne's work.

• In 1876, Édouard Lucas proved that M₁₂₇ was indeed prime, just as Mersenne had claimed. This was the largest known prime number for 75 years!
• In 1883, Ivan Mikheevich Pervushin found that M₆₁ was prime, even though Mersenne thought it was not. This number is sometimes called Pervushin's number.
• Lucas also showed in 1876 that M₆₇ was not prime. But it wasn't until 1903 that Frank Nelson Cole famously showed its factors. He went to a blackboard, calculated 2⁶⁷ − 1, and then multiplied 193,707,721 × 761,838,257,287. Both calculations gave the same huge number: 147,573,952,589,676,412,927. He did all this without saying a word!

It took about 300 years after Mersenne's original list for all the Mersenne primes in that range to be correctly identified and confirmed.

How We Find Mersenne Primes Today

Finding very large prime numbers is a challenging task. However, there are special, fast methods for checking if a Mersenne number is prime. The most effective method is called the Lucas–Lehmer primality test. This test makes it much easier to check Mersenne numbers than other numbers of the same size.

Because of these efficient tests, many of the largest known prime numbers are Mersenne primes. The search for these giant primes has a dedicated following! A lot of computer power is used to find new ones. Much of this work is done through distributed computing, where many people volunteer their computers to work together.

Graph of the number of digits in the largest known Mersenne prime over time, especially in the electronic era.

The search for Mersenne primes changed completely with the invention of electronic computers.

• In 1952, the first Mersenne prime found by a computer was M₅₂₁. It was discovered using the SWAC computer at UCLA. This was the first new Mersenne prime found in 38 years!
• Soon after, the same computer found four more: M₆₀₇, M₁₂₇₉, M₂₂₀₃, and M₂₂₈₁.
• M_4,423 was the first prime with over 1,000 digits.
• M_44,497 was the first with over 10,000 digits.
• M_6,972,593 was the first with over a million digits.

The number of digits in a Mersenne prime M_n is roughly n multiplied by 0.3.

In 2008, mathematicians at UCLA, working with the Great Internet Mersenne Prime Search (GIMPS), found a Mersenne prime with almost 13 million digits. This discovery earned them a $100,000 prize for finding the first known prime with at least 10 million digits.

New Mersenne primes continue to be found:

• On January 25, 2013, Curtis Cooper discovered the 48th Mersenne prime, 2^57,885,161 − 1, which has over 17 million digits.
• On January 19, 2016, Cooper found the 49th Mersenne prime, 2^74,207,281 − 1, with over 22 million digits.
• On January 3, 2018, Jonathan Pace found the 50th Mersenne prime, 2^77,232,917 − 1, with over 23 million digits.
• On December 21, 2018, Patrick Laroche discovered the 51st Mersenne prime, 2^82,589,933 − 1, which has over 24 million digits.
• On October 12, 2024, Luke Durant found the current largest known Mersenne prime, 2^136,279,841 − 1. This number has over 41 million digits and is the first Mersenne prime with an exponent over 100 million!

List of Known Mersenne Primes

As of 2024^[update], the 52 known Mersenne primes are 2^p − 1 for the following p values:

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, 74207281, 77232917, 82589933, 136279841. (sequence A000043 in OEIS)

Factoring Composite Mersenne Numbers

While Mersenne primes are only divisible by 1 and themselves, not all Mersenne numbers are prime. Composite Mersenne numbers can be broken down into smaller prime factors. For example, we saw that M₁₁ = 2¹¹ − 1 = 2047 = 23 × 89. Finding these factors for very large Mersenne numbers is a huge challenge for mathematicians and computers.

Mersenne Numbers in Everyday Life and Puzzles

Mersenne numbers appear in interesting places:

• In the famous puzzle Tower of Hanoi, if you have an n-disc tower, it takes exactly M_n steps to solve it without mistakes.
• The total number of rice grains on a chessboard in the wheat and chessboard problem is M₆₄.
• The asteroid with minor planet number 8191 is named 8191 Mersenne because 8191 is a Mersenne prime.
• In geometry, certain types of right triangles have an inradius (the radius of the largest circle that fits inside the triangle) that is always a Mersenne number. For example, if one leg of a special right triangle is 2^n + 1, its inradius will be 2ⁿ − 1.
• Also in geometry, the number of different shapes (called polytopes) that can be made by cutting the corners of a basic shape and its "dual" (a related shape) is equal to M_n, where n is the dimension of the basic shape. For example, a tesseract (a 4D cube) and its dual have M₄ = 15 different polytopes in their family.

See also

In Spanish: Número primo de Mersenne para niños

• Repunit
• Fermat number
• Power of two
• Mersenne twister
• Great Internet Mersenne Prime Search (GIMPS)
• Largest known prime number
• Perfect numbers