Infinite descent is a method of proof that transforms a hypothetical counterexample into a strictly smaller counterexample, with size measured by a positive integer. Repeating the construction would give
 |
Such a sequence of positive integers is impossible by the well ordering principle. Equivalently, the least possible size of a counterexample would be contradicted by the smaller one (Conrad).
For example, suppose positive integers 
and 
satisfy
 |
Then 
is an even number, and the pair 
satisfies the same equation
with a smaller second coordinate, since 
. This descent proves that 
is an irrational number
without first assuming that 
and 
have no common factor (Conrad,
pp. 1-2).
In Vieta jumping, replacing one integer solution by another can similarly reduce its size. The required positivity and decrease depend on the particular equation.
