How did Fermat use infinite descent to prove an impossibility result?

Pierre de Fermat used the method of infinite descent as a powerful logical tool to prove impossibility results in number theory. This technique serves as an inverted form of mathematical induction.

The strategy follows these steps:

• Assumption: Assume that a set of positive integers exists that satisfies a specific condition or equation.
• Derivation: Use algebraic manipulation to demonstrate that if such a solution exists, it must imply the existence of a smaller set of positive integers that satisfies the same condition.
• Contradiction: By repeatedly applying this argument, one generates an infinite, strictly decreasing sequence of positive integers. Because any set of positive integers must contain a smallest member, such an infinite descent is impossible.
• Conclusion: Since the existence of the initial solution leads to a logical contradiction, one must conclude that no such numbers can exist in the first place.

Fermat notably employed this method to prove that there cannot be a right triangle with sides of integer length whose area is a perfect square.1