In August 1900 at the Collège de France the Second International Congress of Mathematicians was held. Among the attendees was David Hilbert, perhaps the most brilliant mathematician of the 20th century, who gave a lecture entitled * The Problems of Mathematics* . He began by recalling the two most famous mathematical problems that remained to be solved: Fermat’s Last Theorem -solved by the Englishman Andrew Wiles in 1999- and the one known among mathematicians as the three-body problem -which remains unsolvable and possibly don’t have it- He then went on

**to list 23 problems to which no solution was also known,**with the laudable purpose of inspiring future generations of mathematicians. And it didn’t fail. Although 16 remain unsolved, the so-called “Hilbert hit list” has marked mathematics ever since. The mathematician Hermann Weyl said about them: “Anyone who solves even one of them, will enter the honor group of mathematicians.” Of all of them

**the most famous is the so-called Riemann Conjecture**.

This mathematical puzzle formulated in 1859 by the German GFB Riemann has to do with numbers, in particular with the so-called **prime numbers** , those that are only divisible exactly by themselves and by one: 2, 3, 5, 7, 11. .. In other words, they cannot be expressed as the product of two smaller ones.

## The mysterious prime numbers

Prime numbers are quite peculiar and many of the mathematical puzzles that remain today have to do with them. For example, the so-called **Goldbach Conjecture** , stated for the first time on June 7, 1742 in a letter that the mathematician to whom it owes its name addressed one of the greatest mathematicians in history, Leonhard Euler: “Every integer greater than that 5 can be written as the sum of three prime numbers. This conjecture is considered not only **one of the most difficult unsolved problems** in number theory, but in all of mathematics. Euler reformulated it by stating that all even numbers greater than 2 can be expressed as the sum of two primes. Now, is it true? No one has yet been able to prove it, despite the fact that **Faber and Faber offered a prize of one million dollars** for whoever could prove the Conjecture between March 20, 2000 and the same date in 2002 (in 1977 the Russian Pogorzelski claimed to have proved it, but his optimism was not shared by the rest of his colleagues).

Apparently, prime numbers only interest a handful of crazy mathematicians. Surely the only memory we have of them is from our school days, when they gave us factorization exercises in prime numbers. For example, 15 is the product of 3 and 5, which are prime. Factoring numbers is an arduous task and if it weren’t for those rules that we were taught in school it would be quite difficult to achieve our goal. The problem appears when the number to factor is large, like 4294967297. Here all the rules that we were taught fail and if we want to try to factor it we must start doing probatins. With a lot of luck we could discover that this number is the product of 641 times 6700417, which are primes.

## Looking for a pattern among the cousins

You may be wondering why we have chosen this number. Because it is **the fifth number of Fermat** , a 17th century mathematician who invented a very simple series that is constructed as follows: take two and raise it to two raised to a natural number. In our case, since it is the fifth Fermat number, it rises to the fifth. And at the end one is added. Our fifth number is therefore two to the power of two to the power of five plus one, or what is the same, 2 to the power of 32 and what comes up plus one. The curious thing about this series of gigantic numbers is that Fermat believed that it was a series of prime numbers. But he was wrong because in 1732 the great Euler factored the fifth number. Little by little, the following were factored: the eighth fell in 1980 and the ninth in 1990, in a challenge that required several weeks of collaboration between 200 mathematicians and a thousand computers around the world.

The search for ever larger prime numbers is one of the great mathematical challenges. The last prime, the largest known to date, was found on December 7, 2018 and **has 23249425 digits** : it is 2 ^{82589933} − 1. It was discovered by Patrick Laroche, a Florida computer scientist who belongs to the GIMPS (Great Search for Numbers) group. Mersenne Primes on the Internet) that since 1996 has been dedicated to precisely that, looking for prime numbers of the style 2 raised to a number minus 1.

The search for the next prime number does not stop. And not just for the itch of glory, but also because the Electronic Frontiers Foundation is offering **a $250,000 reward to whoever finds the first billion-digit prime** .

## Cousins and the security of nations and banks

Now we might ask, as many high school students do, what is the use of wasting time on these things? The reason is simple: if someone discovers a quick way to factor large numbers, they **could bypass the security checks of** banks, credit cards, Ministries of Defense… To keep their data hidden from possible curious, they encrypt it with codes based on the prime factors of large numbers.

And this is where we link to the Riemann Conjecture and its importance for both pure and applied mathematics. The German mathematician wondered if the distribution of prime numbers along the sequence of natural numbers follows any pattern. In other words, **is there any way to predict where we are going to find a prime number?** Riemann’s bet 145 years ago is that the frequency of occurrence of prime numbers is related to all the interesting solutions of an elaborate mathematical function then called **Riemann’s Zeta function** . So far it has not been possible to prove whether it is true, something that is especially unnerving considering that a considerable number of important mathematical theorems depend on this hypothesis being true.

*References:*

*Mazxur, B. (2016) Prime Numbers and the Riemann Hypothesis, Cambridge University Press*