Proof by Infinite Descent

A proof by infinite descent, also called Fermat's method of descent, is a kind of proof by contradiction used to show that a statement cannot hold for any natural number: the method shows that if the statement held for some number, it would also have to hold for a smaller number, and repeating this indefinitely produces an infinite decreasing sequence of natural numbers, which is impossible, giving a contradiction. It is often used to show that a given equation, such as a Diophantine equation, has no solutions, typically by assuming a solution exists and deriving a second, smaller solution, or equivalently by assuming a smallest counterexample and deriving a still smaller one. The technique traces back to Euclid's Elements, in Proposition 31 of Book 7, but it takes its common name from Pierre de Fermat, who developed and popularized it, applying it extensively to Diophantine equations. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/