Vieta jumping and small norms in quadratic number fields

In this article we explain the connection between the famous problemfrom the IMO 1988 and elements of small norms in quadratic number fields with parametrized units.

💡 Research Summary

The paper by Franz Lemmermeyer revisits the celebrated 1988 International Mathematical Olympiad problem which asks to prove that for positive integers a, b, if ab + 1 divides a² + b² then the quotient (a² + b²)/(ab + 1) is a perfect square. The author first rewrites the condition as a² + b² = k(ab + 1) and observes that the pair (a,b) lies on the conic Cₖ: x² − kxy + y² = k. The classical “Vieta jumping’’ technique is introduced as an operation that, given a point on Cₖ, replaces one coordinate by the other root of the quadratic equation obtained by fixing the second coordinate. Explicitly the two involutions are
 ♭(a,b) = (k a − b, b) and ♯(a,b) = (a, k b − a).
Iterating these involutions generates two infinite sequences of integral points, described by the recurrence aₙ₊₂ = k²aₙ₊₁ − aₙ and the identity aₙaₙ₊₂ = aₙ₊₁² − k². Starting from the obvious points (±k,0) and (0,±k) every integral solution of the original equation is shown to belong to one of these sequences; this gives a complete elementary description of all solutions.

The novelty of the paper lies in interpreting Vieta jumping as the action of units in a real quadratic number field. By completing the square, Cₖ can be written as (x − κy)² − (κ² − 1)y² = k with κ = k/2 (κ may be half‑integer). The map (a,b) ↦ α = a − κb + b√m, where m = κ² − 1, identifies points on Cₖ with elements of the field K = ℚ(√m) of norm N(α) = k. The fundamental unit ε = κ + √m has norm +1, and multiplication by ε corresponds exactly to the involution ♭: αε = (k a − b) + a√m. Thus Vieta jumping is nothing but repeated multiplication by the unit ε (or its inverse), and the set of integral points on Cₖ is a principal homogeneous space for the Pell conic associated with ε. This algebraic viewpoint explains why the descent process works and provides a conceptual bridge between Olympiad‑level combinatorial arguments and algebraic number theory.

Using this bridge, the author proves several general results about Diophantine equations of the form x² − pxy + y² = q. Theorem 6 shows that for integers p > 2 and 0 < q ≤ p + 1, the existence of an integral solution forces q to be a perfect square; the bound is sharp because when q = p + 2 there are infinitely many solutions generated by Vieta jumping. The proof proceeds by repeatedly applying the unit action to reduce any solution to a bounded region near the origin; the only points that survive this reduction are those with q a square. Theorem 7 treats the case 3 − p ≤ q < 0, showing that no solutions exist except for the special value q = 2 − p, where the solutions are described explicitly in terms of Fibonacci numbers. Further propositions explore related families, such as the equation 3xy − 1 | x² + y², whose solutions are again parametrized by Fibonacci numbers.

In the final section the paper returns to the number‑field perspective to address “small norm’’ elements. Proposition 11 (a refined version of a classical result of Dirichlet and Chebyshev) asserts that for any ξ in a real quadratic field K with norm ν, there exists a power εʲ of a fundamental unit ε such that ξεʲ = a + b√m with |a| and |b| bounded by constants depending only on ν and ε. This quantitative control of the coefficients is precisely what is needed to guarantee that the Vieta‑jumping descent terminates after finitely many steps, and it also yields an algorithmic method for finding all elements of norm k in the field.

Overall, Lemmermeyer’s article demonstrates that Vieta jumping is not an isolated Olympiad trick but a manifestation of the action of units on a principal homogeneous space attached to a Pell conic. By embedding the problem into the arithmetic of quadratic number fields, the paper provides a unified framework that simultaneously yields elementary proofs of classical Olympiad results, generalizations to broader families of Diophantine equations, and new insights into the distribution of small‑norm elements in real quadratic fields.