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This is a Quantitative Study study.

October 12, 2025Open Access

Diophantine tuples and Integral Ideals of Q (d)

KCKalyan ChakrabortyLamar University SGShubham GuptaMedanta The Medicity KKKrishnarjun KrishnamoorthyChinese Mathematical Society

Key Points

The study establishes a link between diophantine tuples and integral ideals in quadratic extensions.
For every diophantine m-tuple with property D(n), integral ideals can be constructed using the norm map.
The construction of ideals is explicitly demonstrated for diophantine pairs where the GCD condition holds.
This research enhances understanding of the integral ideals in relation to diophantine equations.

Abstract

Suppose n is the fundamental discriminant associated with a quadratic extension of Q. We show that for every Diophantine m-tuple \t₁, t₂, , tₘ\ with the property D (n), there exists integral ideals t₁, t₂, , tₘ of Q (n) and c \1, 2\ such that tᵢ= cN (tᵢ) for i=1, 2, , m. Here, N () denotes the norm map from Q (n) to Q. Moreover, we explicitly construct the above ideals for Diophantine pairs \a₁, a₂\ whenever (a₁, a₂) = 1.

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