Prime numbers and the Golden Theorem – 12/12/2023 – Marcelo Viana

This is a story about the mysteries of mathematics, about how things that seem to have nothing to do with each other turn out to be intimately related. It starts with the odd prime numbers: 3, 5, 7, 11, 13, 17, 19, 23, …. They can be separated into two types: the 4k+1 type, formed by those (such as 5, 13, 17, …) whose division by 4 gives remainder 1; and the 4k+3 type, consisting of those (7, 11, 19, 23, …) whose division by 4 gives remainder 3.

At first glance there is nothing notable that distinguishes one type from another. But in his book “Disquisitiones Arithmeticae” (Arithmetic Investigations), published in 1801, Gauss pointed out a very interesting mathematical fact, which I will try to explain.

Consider any two odd primes p and q. We say that p is a perfect square modulo q if there exists some integer n such that nxn – p is a multiple of q. For example, 5 is a perfect square modulo 19: just consider n=9 and note that 9×9-5=76 is a multiple of 19.

Interestingly, we can reverse the roles of the two primes: 19 is also a perfect square modulo 5. To see this, take n=8 and note that 8×8-19=45 is a multiple of 5. We say that 5 and 19 are reciprocal primes. Now, Gauss showed that two odd primes can only be reciprocal if at least one is of the type 4k+1. Who would say?!

Gauss was so excited by this result that he called it the Golden Theorem. It is currently known as the Quadratic Reciprocity Theorem. In fact, Euler had discovered this theorem before, but was unable to prove it. And Legendre gave a proof, but it was incomplete. The first correct proof was given by Gauss, who, in fact, published six different proofs (two more were found in his unpublished papers!). Today, more than 250 different tests are known.

Precisely, the Quadratic Reciprocity Theorem says that if any of the primes p and q are of the type 4k+1 then it is all or nothing: either they are reciprocal (this is the case of 5 and 19, as we have seen), or none of them is a square perfect module the other (as with 5 and 13, for example). If they are both of the 4k+3 type, then exactly one of them is perfect quadratic modulo the other. For example, 11 is a perfect square modulo 7, but 7 is not a perfect square modulo 11 (check!). In particular, in this case the two primes are never reciprocal.

Like every great theorem, this one raises many other questions. Is there a cubic reciprocity theorem (using cubes instead of squares)? And with any other power? That’s the subject for a course at Impa!

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