In this short introductory article, I will take you on a little journey into the exotic world of prime numbers.
It all started in France with the birth of a boy, that was to become one of the greatest scientists, philosophers, and mathematicians of his time.
Marin Mersenne was born on 8 September 1588. He was a priest, scientist, and mathematician studying everything of interest from vibrating strings on musical instruments to different fields of mathematics.
Mersenne was in a sense a center of science at his time corresponding with Galileo Galilei, René Descartes, Étienne Pascal, Pierre de Fermat, among other fellow scientists and philosophers of the time.
Even though he was not only a mathematician, he is most famous for his work of certain types of prime numbers now called Mersenne primes in his honor. These prime numbers had actually been studied almost 2000 years before Mersenne was born, but he compiled a list of some of them and this list became very famous.
People tried to prove that the numbers on the list were prime numbers (some were actually not) but the task was daunting due to the size of the numbers.
Mersenne claimed for instance that 2¹²⁷ — 1 is a prime number. In 1876 Édouard Lucas proved this to be correct. It would be 75 years before anyone found a bigger one and it is still the largest prime found by hand ever.
The Geometric Sum
Suppose that we have a sum of the form
s(x) = 1 + x + x² + x³ + ⋅⋅⋅ x^(n-1)
You might know that there is a little trick to calculate this.
Before moving on, let’s define a Mersenne prime.
Of course, not all numbers on that form are prime numbers but it turns out that the following is true.
Let’s prove that theorem by proving the contrapositive (and therefore equivalent) statement
Assume that n is a composite number. Then we can write n = ab for some a, b ∈ ℕ with a, b > 1. By the closed form of the power sum above we now have:
which of course proves the theorem.
A number of this form, prime or not, is called a Mersenne number.
Note that the converse statement is not true. You can have a Mersenne number with prime exponent, which is composite e.g. 2¹¹ - 1 = 23 ⋅ 89.
This theorem suggests that looking for large primes would be more efficient if we restricted our search to the Mersenne numbers with prime exponent and this actually turns out to be true. However, Mersenne primes become very rare among Mersenne numbers when the exponent becomes large.
A Deep Mystery
Mersenne didn’t leave any evidence as to how he came up with the numbers on the list, nevertheless, the hunt for primes had started and the search for these shy numbers is still going to this day!
The largest primes we know of today are Mersenne primes and large primes play a critical role in cyber-security and cryptography which is the science of encoding and decoding information, and many of its algorithms, such as RSA, rely heavily on prime numbers, therefore these numbers are of importance in our modern society even though Mersenne himself would never have thought of that.
Mersenne came up with a list that, thanks to modern computers, have now grown considerably, but he could not prove if there are infinitely many Mersenne primes, neither could Euler! And guess what. Neither can anybody.
This is still an open problem. A mystery of numbers.
It turns out that there is a connection between Mersenne primes and certain other types of mysterious numbers called perfect numbers, that is exactly what I will explore together with you in my next article.
See you then.