Infinite Descent & Vieta Jumping

No smallest counterexample can survive.

Assume a solution exists, take the smallest one, and construct a strictly smaller one. Since the positive integers cannot descend forever, no solution existed. Fermat's original weapon, and still the sharpest for 'prove there are no solutions'.

Vieta jumping is descent wearing a quadratic disguise. Fix the value $k$ in a symmetric condition, read it as a quadratic in one variable, and jump from one root to the other — Vieta's formulas guarantee the partner is an integer, and it is smaller.

IMO 1988/6 is the archetype: $ab+1 \mid a^2+b^2$ forces the quotient to be a perfect square, proved by descending from a minimal counterexample.

Where it appears on Lemma: Level 5 (Diophantine equations) and Level 8 (the IMO set).

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Infinite Descent & Vieta Jumping unlocks at Level 6 of Lemma's eight-level ladder, with lessons that teach it and drills that make it stick. 78 lessons, 624 curated problems and unlimited generated practice at six difficulties. Free to start — no card, no trial clock.

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