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.

More techniques

• Algorithmic & Greedy ConstructionsDescribe a procedure, then prove it terminates and works.
• The Extremal Principle in GraphsTake the longest path, the vertex of largest degree.
• All 29 techniques →