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


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*Answers, each of which outlines a fundamentally (or at least demonstrably) different mechanism which comprises the critical step(s) in an ID proof.

*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?

Parity arguments plus substitution and elimination of common factors.
Example: The classical proof that $\sqrt{2}$ is irrational uses this mechanism.
