A perfect square from IMO 1988

Unsolved Very hard Archive

archive posted 5 weeks ago

Let $a$ and $b$ be positive integers such that $ab + 1$ divides $a^{2} + b^{2}$ . Prove that

\frac{a ^{2} + b ^{2}}{ab + 1}

is the square of an integer.

Posed by International Mathematical Olympiad, problem 6, 1988

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History and references

It was the last problem of the 1988 olympiad in Canberra and is often called one of the hardest olympiad problems of its time. The standard proof, now known as Vieta jumping, takes a pair with $a + b$ as small as possible for a given value $k$ of the quotient, treats $a$ as a root of the quadratic $x^{2} - kb x + b^{2} - k = 0$ , and uses the other root to build a smaller pair, unless $k$ is a perfect square.

References

Vieta jumping

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