Why are prime numbers so important?

Natural numbers are those we use for counting: 1, 2, 3, 4… A prime number is a natural number greater than 1 that has only two positive divisors: itself and 1. So, 2 is a prime number because it can only be divided by 2 and 1 with a whole number. 3 is also a prime number for the same reason. But 4 is not a prime number because it can be divided by 1, 2, and 4. There are infinitely many prime numbers, as the Greek mathematician Euclid proved.

As you say in your question, they are very important. And they are important from different points of view. The first is focused on mathematics itself because these numbers are the backbone of many branches of this science; for example, they are fundamental to number theory.

But they are also a good example of how mathematics evolves. Initially, they began to be studied with the sole aim of increasing knowledge. That is, prime numbers weren't investigated to find applications but simply out of curiosity to understand them better. Researchers investigated their properties, whether there was a formula to calculate prime numbers, how many prime numbers there were less than or equal to a given number, whether there was a pattern to identify them, how to determine if a large natural number was prime, and so on. The best mathematicians have investigated them. I've already mentioned Euclid, known as the father of geometry, but also other Greek mathematicians like Eratosthenes; and more recently, Pierre de Fermat, Leonhard Euler, Gottfried Wilhelm Leibniz, and others. Leibniz, Sophie Germaine, and Carl Friedrich Gauss , among many others. The Riemann Hypothesis, still unproven, one of theClay Mathematics Institute 's Millennium Problems with a one million dollar prize for its solution, is closely related to prime numbers.

In basic research, which is done to expand knowledge rather than with an immediate application in mind, most of the time, once we have that knowledge, an application is found. And this often happens in mathematics, as has been the case with prime numbers. For example, thanks to them, there was a major revolution in cryptography, the science of encrypting or encoding messages to make them inaccessible to unauthorized users, and which is key for the internet and the applications we use to function correctly. So we find that the properties of prime numbers have paved the way for such important technologies as communications, email, and e-commerce, among many others.

One of the key properties of prime numbers for these applications is that they don't follow a pattern. Their distribution seems unpredictable, random. That's why programs are used to search for new prime numbers. Some of these searches request citizen collaboration; that is, anyone can lend their computer to be used, along with thousands or hundreds of thousands of others, in the search.

The prime number with the most digits found so far has more than 41 million digits and was found thanks to one of these citizen collaboration programs in which volunteers from all countries participate.

There are some other issues related to prime numbers that are really interesting. For example, I work with population models. And it turns out that prime numbers appear in certain insect life cycles when studying populations. There are some cicadas that have life cycles of 13 or 17 years, both prime numbers . These insects live underground and every so often, every 13 or 17 years, they emerge to the surface for a few days to reproduce. The explanation given is that this gives them an evolutionary advantage, as argued by paleontologist Stephen Jay Gould in his work *Of Bamboos, Cicadas, and the Economy of Adam Smith *. If their life cycles were non-prime numbers—6, 8, 10, etc.—they would coincide with many more of their predators, which have short and regular life cycles.

And there's something funny about croquettes. It seems prime numbers have even made their way into marketing. Often, when you order a plate of croquettes, they give you a prime number, usually 5, because most of the time that prime number can't be divided equally among the people who ordered them (unless, of course, it's 1 or 5), which forces you to order another plate. Another option, of course, is to divide them diplomatically, but that takes away some of the excitement of the little mathematical dilemma of the appetizer.

Victoria Otero Espinar is a professor of Mathematical Analysis in the Department of Statistics, Mathematical Analysis and Optimization at the University of Santiago de Compostela and a researcher at the Center for Research and Mathematical Technology of Galicia (CITMAga), she is also president of the Royal Spanish Mathematical Society.

Coordination and writing: Victoria Toro .

Question submitted via email by Carla Gómez Inaraja .

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