29/11/2023

Folha: 'Prime numbers and the curious way of doing calculations'

Reprodução

Reproduction of Marcelo Viana's column in Folha de S. Paulo.

Prime numbers are among the simplest and most mysterious concepts in mathematics . The definition is simple: an integer n greater than 1 is prime if it has only two divisors, itself and 1. But the way primes are distributed among all integers still holds many mysteries.

An easy observation is that there are three types of primes. We have 2, which is the only even prime. Then come primes of the form 4k+1, whose division by 4 gives a remainder of 1. Finally, there are primes of the form 4k+3, whose division by 4 gives a remainder of 3. For example, 5 and 13 belong to the second type, while 7 and 11 belong to the third.

At first glance, these last two types of primes are very similar, and nothing seems to indicate that one is more special than the other. However, surprisingly, that is precisely the case. And the proof is based on a mathematical result so special that the great Gauss called it the "golden theorem".

To explain, I need to talk about a curious way of doing calculations that mathematicians call modular arithmetic. It works like this: Initially, we fix an integer n greater than 1, called the modulus. In the calculations, only the numbers 0, 1, 2, … n-1 are used. To add two of them, we add them in the usual way, but we take as the result the remainder of the division of that sum by n. To multiply, we do the same: we multiply in the usual way and take as the result the remainder of the division of that product by n.

We learn in school that an integer is called a perfect square if it is the square of another integer. For example, 49 is a perfect square because it is equal to 7 x 7. This notion also makes sense in modular arithmetic, using the respective multiplication. For example, 13 is a perfect square modulo 17, because 13 = 8 x 8 (modulo 17).

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