Vieta jumping and small norms in quadratic number fields

TL;DR

The paper links a classical IMO problem to the arithmetic of real quadratic fields by combining Vieta jumping with Pell-conic dynamics and norm considerations. It shows that the condition $(a^2+b^2)/(ab+1)$ being an integer forces the parameter $k$ to be a square, using descent on conics and a family of Diophantine equations. Beyond the original problem, it proves general results on $x^2 - pxy + y^2 = q$, connects integral points to Pell equations, and develops a quadratic-number-field framework that yields bounds and corollaries for small-norm elements. The work demonstrates how olympiad-style techniques can be enriched by algebraic number theory to obtain multiple independent proofs and broad generalizations with potential applications to related Diophantine problems.

Abstract

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.

Figures (5)

• Figure 1: Vieta jumping on ${\mathcal{C}}_4: x^2 - 4xy + y^2 = 4$.
• Figure 2: Vieta jumping on ${\mathcal{C}}_m: a^2 - m^2 ab + b^2 = m^2$ for points on the lower (left) and upper branch (right).
• Figure 3: Vieta jumping on $x^2 - 3xy + y^2 = -1$ in the first quadrant.
• Figure 4: Integral points on $x^2 - xy + y^2 = 1$
• Figure 5: Vieta jumping on the Pell conic ${\mathcal{P}}: x^2 - 2xy - y^2 = 1$ and on ${\mathcal{P}}^{-}: x^2 - 2xy - y^2 = -1$.

Theorems & Definitions (26)

• Proposition 1
• Proposition 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Theorem 6
• proof : Proof fo Thm. \ref{['TM1']}