How did Pierre de Fermat solve the case n = 4 related to Fermat’s last theorem?

Pierre de Fermat did not explicitly write out a proof for the general case of his "last theorem," but he successfully demonstrated the case for $n=4$ by proving a related theorem using his method of infinite descent.12

This technique is a form of mathematical induction used to prove the impossibility of a statement. The logical strategy involves assuming that a set of positive integers exists that satisfies a specific condition, and then using that assumption to generate a smaller set of positive integers that also satisfies the same condition. By repeating this process, one would produce an infinitely decreasing sequence of positive integers. Because any set of positive integers must contain a smallest member, this sequence cannot exist, creating a contradiction that proves no such numbers could have existed in the first place.32