Method of Infinite Descent

Intro to Proofs · Axiom Academy

LESSON Method of Infinite Descent Fermat's trick for proving something cannot exist: turn one solution into a strictly smaller one — forever — and let the positive integers refuse to fall that far. Infinite descent is proof by contradiction with a twist. Suppose the thing we want to disprove exists. We then show that from any such object we can build a strictly smaller one with the same property. Track its "size" — a positive integer like z_0 — and that step produces z_1 < z_0 , then z_2 < z_1 , and so on without end. Each step makes the size strictly smaller But every z_k is a positive integer — there is a floor 2. Worked Example: x^2 + y^2 = 3z^2 Let's prove there are no positive integers x, y, z with x^2 + y^2 = 3z^2 . Suppose (x_0, y_0, z_0) is a solution. We will manufacture a smaller one, (x_1, y_1, z_1) with z_1 < z_0 . Every square is or . So only when both and . Write x = 3x_1 and y = 3y_1 . Then 9x_1^2 + 9y_1^2 = 3z^2 . Dividing by 3 gives 3(x_1^2 + y_1^2) = z^2 , so , hence . Write z = 3z_1 . Substituting yields x_1^2 + y_1^2 = 3z_1^2 with z_1 = z_0/3 < z_0 — same equation, smaller z . The new triple is again a positive-integer solution, so the same argument applies to it, producing through the positive integers — impossible. Therefore no solution exists at all.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.