What are the principal (different) mechanisms of infinite descent proof?

Question

I’m interested in building a list (including, where possible, links to proofs/papers/examples) which presents all known mechanisms of infinite descent (ID).

I think this list would best be presented in two parts:

1. Answers, each of which outlines a fundamentally (or at least demonstrably) different mechanism which comprises the critical step(s) in an ID proof.
2. Comments to each answer, giving concrete examples of that type of ID proof.

I’ve started by giving several [community wiki] answers myself, in roughly the form I think will be helpful to future readers.

Now… Do any ID proofs use the fact that every odd number is the sum of the squares of four integers which sum to 1? Or Bezout’s identity? etc. What are all of the methods and mechanisms?

Answer 1

Parameterization of primitive Pythagorean triples, with multiplication of the resulting components.

Example: Fermat’s proof that the equation $X^4-Y^4=Z^2$ has no non-trivial integer solutions uses this mechanism.

Answer 2

Parity arguments plus substitution and elimination of common factors.

Example: The classical proof that $\sqrt{2}$ is irrational uses this mechanism.

Answer 3

Parameterization of primitive Pythagorean triples, with a comparison of two different factorizations of the same number.

Example: Van der Poorten’s proof of Fermat’s theorem regarding the impossibility of having four integer squares in arithmetical progression uses this mechanism.

Answer 4

Standard Vieta Jumping.

The minimal solution $(A, B)$ with respect to some function of $A$ and $B$, usually $A+B$, is taken. The equation is then rearranged into a quadratic with coefficients in terms of $B$, one of whose roots is $A$, and Vieta's formulas are used to determine the other root to the quadratic. See this Wikipedia entry for more details.