## Diophantine Quintuples over Quadratic Rings

**Authors:** Philip Gibbs

A Diophantine m-tuple is a set of m distinct non-zero integers such that the product of any two elements of the set is one less than a square. The definition can be generalised to any commutative ring. A computational search is undertaken to find Diophantine 5-tuples (quintuples) over the ring of quadratic integers Z[√D] for small positive and negative D. Examples are found for all positive square-free D up to 22, but none are found for the complex rings including the Gaussian integers.

**Comments:** 11 Pages.

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### Submission history

[v1] 2018-02-14 10:10:56

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