Certain Diophantine Tuples in Imaginary Quadratic Fields

2020/03/06 by Gupta, Shubham
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2003.03298

Abstract

Let K be an imaginary quadratic field and O_K be its ring of integers. A set \a₁, a₂, ⋯,a_m\ ⊂ O_K∖\0\ is called a Diophantine m-tuple in O_K with D(-1) if a_ia_j -1 = x_ij², where x_ij ∈ O_K for all i,j such that 1 ≤ i < j ≤ m. Here we prove the non-existence of Diophantine m-tuples in O_K with D(-1) for m > 36.

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