Infinite Descent

Build a smaller counterexample; the positive integers can't descend forever.

The idea

Infinite descent proves that an equation has no solutions in whole numbers, using one structural fact: a strictly decreasing sequence of positive integers cannot continue forever. Reach for it when a problem asks you to show that an equation has no nontrivial integer solution, or that a number is irrational, and scaling all the variables by a common factor leaves the equation unchanged.

The argument has a fixed shape. Assume a solution exists, and among all solutions choose one minimizing some positive integer quantity: the smallest $|x|$, the smallest $z$, a fraction in lowest terms. Then use the equation to construct another solution of the same form in which that quantity is strictly smaller. This contradicts the minimality of the chosen solution, so no solution exists.

The smaller solution nearly always comes from divisibility. A congruence argument shows that some fixed prime must divide every variable; dividing each variable by that prime returns the original equation in smaller numbers, which is the descent step.

Ways to work on it

• Walkthrough. Show an equation has no solutions by descending from a supposed smallest one.
• Practice. Spot the contradiction a descent step produces.
• Hardest. Rule out nonzero solutions of a cubic equation by descent.

Not sure where to start? Take the ten-question placement test.