Congruence obstructions for D(z)-quadruples in ℤ[√(-2)]

Abstract

Let ω=√(-2). A D(z)-m-tuple in ℤ[ω] is a set of m nonzero elements whose pairwise products, increased by z, are all squares. Dujella and Soldo determined the z for which a D(z)-quadruple exists, up to three congruence classes and the three individual values z ∈ {-1, 1 ± 2ω}; for z=-1 they record the conjecture that none exists. Both non-existence statements in that classification follow from a congruence argument. We show that the method is exhausted. If z is a difference of two squares of ℤ[ω] — by a theorem of Dujella and Franušić, exactly when z avoids the two classes already excluded — then for every modulus m ≥ 1 there are four distinct nonzero elements of ℤ[ω] all six of whose pairwise products, increased by z, are squares modulo m. Consequently no congruence argument at any modulus can prove that no D(-1)-quadruple exists, and none can settle any remaining case of the classification negatively. A congruence can still obstruct the extension of a fixed triple, and we exhibit an infinite family of such triples at z=-1: for b ∈ ℤ the set T_b={-2, b²-1-bω, b²-1+bω} is a D(-1)-triple whenever b ≠ 0, and whenever 8 ∤ b it lies in no D(-1)-quadruple, by a single reduction modulo 16. Two of its three entries are not rational integers. All statements are formally verified in Lean 4, apart from two small steps that are identified in the text as done by hand.

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