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.

Answer 5

Showing that a function $F: P \to P$ has the property that for every $z \in P$ there exist a positive $k$ such that

$\quad F^k(z) \text{ is a fixed point of } P$

by associating to each $z \in P$ an integer $m$ that decreases (when not $z$ is not a fixed point) as $F$ is applied.

Note: In the third paragraph of the wikipedia article on mathematical induction you'll find this sentence

Mathematical induction, in some form, is the foundation of all correctness proofs for computer programs.

Example: Termination of Euclid's GCD algorithm. (see proposition 3 from this answer).

Answer 6

Set theoretic elementary result:

Every subset of a finite set is finite.
(for definition of finite and argument see this math.stackexhange post).

Keep decrementing a positive integer $n$ by $1$.

Answer 7

Factoring until the logic prism (predicate) shatters as you crash through the ground floor.

You can prove Euclid's lemma using infinite descent.

See this link.

Answer 8

Number theory 101

Descending to a factor of $A$ in $A + B$.
Removing $A \gt 0$ in $A + B$ but if $A = 0$ descend to a factor of $B$.
Decrementing by $1$ a factor of $A$ in $A + B$

The theory of both Euclidean division (over $\Bbb N$) and Base-$\text{b}$ representations can be handled using these mechanisms.

Answer 9

Fundamental Theorem of Arithmetic (uniqueness)

Reducing a multiplicand of an integer $n$.

See

https://math.stackexchange.com/a/3309478/432081