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On \(D(-1)\)-triples \(\{1,4p^2+1,1-p\}\) in the ring \({{\mathbb {Z}}}[\sqrt{-p}]\) with a prime p

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Abstract

Let p be a prime such that \(4p^2+1\) is also a prime. In this paper, we prove that the \(D(-1)\)-set \(\{1,4p^2+1,1-p\}\) cannot be extended with the forth element d such that the product of any two distinct elements of the new set decreased by 1 is a square in the ring \({{\mathbb {Z}}}[\sqrt{-p}]\).

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Acknowledgements

Authors would like to thank the anonymous referee for helpful remarks and suggestions which improved the first version of the paper.

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Correspondence to Ivan Soldo.

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The first and the third author was supported by the Croatian Science Foundation under the project no. IP-2018-01-1313.

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Filipin, A., Bokun, M.J. & Soldo, I. On \(D(-1)\)-triples \(\{1,4p^2+1,1-p\}\) in the ring \({{\mathbb {Z}}}[\sqrt{-p}]\) with a prime p. Period Math Hung 85, 292–302 (2022). https://doi.org/10.1007/s10998-021-00435-5

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