Infinite descent

Infinite descent is a proof technique that rules out a proposed solution by showing it would generate an indefinitely decreasing sequence of positive integers, which cannot exist.

Minimal counterexample: A proof strategy that assumes a counterexample exists and selects one minimal under a chosen measure. Descent contradicts the existence of a counterexample chosen with minimal size.
Fermat's right-triangle theorem: The theorem that no integer square is the sum of two positive fourth powers. Fermat’s descent proves this by producing a smaller right triangle of the same forbidden kind.
Pierre de Fermat: A seventeenth-century French mathematician known for contributions to number theory and analytic geometry. Fermat explicitly described infinite descent and used it in several number-theoretic arguments.
Positive integer: An integer greater than zero. The descent measure must take positive-integer values so it has a least possible value.
Strong induction: An induction method in which a case may rely on all smaller cases, not just the immediately preceding one. Both methods exploit the positive integers’ ordering, but descent argues backward from a hypothetical failure.
Fermat's last theorem: The theorem that no positive integers satisfy xⁿ + yⁿ = zⁿ for any integer n greater than two. Fermat claimed a descent proof for the exponent-four case, a central historical connection.
Diophantus: An ancient Greek mathematician whose work on equations survives chiefly in the Arithmetica. Fermat’s descent arose in engagement with Diophantine problems and readings of the Arithmetica.
Well-ordering principle: Every nonempty set of positive integers has a least element. This principle supplies the minimum that an infinite descent would contradict.
Fermat's method of infinite descent: Pierre de Fermat’s proof method that derives an endless decrease among positive integers from an assumed solution. This named method gives the technique its historical form and a classic application.
Pythagorean triple: A triple of positive integers whose first two squares sum to the square of the third. The classic descent constructs a smaller primitive triple while preserving the required square conditions.

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Legendre's three-square theoremRelated: Descent arguments help establish that integers outside the excluded family admit a representation.