Prime numbers and a curious way of doing math – 11/28/2023 – Marcelo Viana

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

An easy observation is that there are three types of primes. We have 2, which is the only even prime. Then come the primes of the form 4k+1, whose division by 4 gives a remainder of 1. Finally there are the 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 cousins ​​are very similar and nothing seems to indicate that one of them is more special than the other. However, surprisingly, this is exactly what happens. 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 math that mathematicians call modular arithmetic. It works like this. Initially, we fix an integer n greater than 1, called modulo. Only the numbers 0, 1, 2, … n-1 are used in the accounts. To add two of them, we add them in the usual way, but we take as the result the remainder of dividing that sum by n. To multiply, we do the same: we multiply in the usual way and take as a result the remainder of dividing this product by n.

For example, suppose n=7. Then 5+6=4 (modulo 7) because the usual sum of 5 and 6 is 11, and the remainder from dividing 11 by n=7 is 4. Analogously, 5×6=2 (modulo 7) because the usual product of 5 by 6 is 30, and the remainder of dividing 30 by n=7 is 2. This multiplication can have slightly strange properties: note, for example, that 3×8=0 (modulo 12).

We learned 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 7×7. This notion also makes sense in modular arithmetic, using the respective multiplication. For example, 13 is a perfect square modulo 17, because 13=8×8 (modulo 17).

A curious thing is that the roles of 13 and 17 in this statement can be interchanged: 17 is also a perfect square modulo 13 (check!). Therefore, we say that 13 and 17 are reciprocal primes. It turns out that when it comes to reciprocity, the two types of odd primes, 4k+1 and 4k+3, have completely different properties. Want to know how? So don’t miss next week’s column!

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