On the extendibility of certain $$D(-1)$$ D ( - 1 ) -pairs in imaginary quadratic rings

Yasutsugu Fujita () and Ivan Soldo ()
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Yasutsugu Fujita: Nihon University
Ivan Soldo: University of Osijek

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 1, 156-162

Abstract: Abstract Let R be a commutative ring with unity 1. A set of m different non-zero elements in R such that the product of any two distinct elements decreased by 1 is a perfect square in R is called a $$D(-1)$$ D ( - 1 ) -m-tuple in R. In the ring $$\mathbb {Z}[{\sqrt{-k}}]$$ Z [ - k ] , with an integer $$k\ge 2$$ k ≥ 2 , we consider the $$D(-1)$$ D ( - 1 ) -pairs $$\{a,2^i p^j\}$$ { a , 2 i p j } , where $$i\in \{0,1\}$$ i ∈ { 0 , 1 } , a, j are positive integers, p is an odd prime, $$\gcd (a, 2^i p^j)=1$$ gcd ( a , 2 i p j ) = 1 and $$a 1$$ a > 1 .

Keywords: Diophantine m-tuple; Pell equation; 11D09; 11D45 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-023-00465-4

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