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The regularity conjecture for D(4)-quadruples

Let a D(4)D(4)-quadruple be a set of four distinct positive integers such that the product of any two distinct elements, increased by 44, is a perfect square. A regular D(4)D(4)-quadruple is one obtained from a D(4)D(4)-triple {a,b,c}\{a,b,c\} by adjoining

d±=a+b+c+12(abc±(ab+4)(ac+4)(bc+4)).d_{\pm}=a+b+c+\frac{1}{2}\left(abc\pm\sqrt{(ab+4)(ac+4)(bc+4)}\right).

Regularity conjecture. Any D(4)D(4)-quadruple is regular.

Regular quadruples are explicitly generated from triples, while an irregular quadruple would be one not arising in this way. The paper studies this conjecture for pairs of the form {a,ka}\{a,ka\} with k∈{2,3,6}k\in\{2,3,6\} and proves that every D(4)D(4)-quadruple containing such a pair is regular; the general case remains open.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Regularity conjecture for D(4)-quadruples

    Let {a,b,c,d}\{a,b,c,d\} be a D(4)D(4)-quadruple, meaning a set of four distinct positive integers such that the product of any two distinct elements increased by 44 is a perfect square. A D(4)D(4)-quadruple is regular if it is obtained as one of the regular quadruples constructed from a D(4)D(4)-triple by adjoining an element d+d_{+} or, when nonzero, d−d_{-}. Regularity conjecture. Any D(4)D(4)-quadruple is regular. The analogous uniqueness questions for extending triples remain open, while the nonexistence of D(4)D(4)-quintuples is known.

    source: Marija Bliznac Trebješanin, “Extension of a Diophantine triple with the property D(4)”, arXiv:1906.03914 (2020).

Sources & referencesView supporting material

Primary source

Kouèssi Norbert Adédji, Marija Bliznac Trebješanin, Alan Filipin and Alain Togbé, “On the D(4)-pairs \a, ka\ with k\2,3,6\”, arXiv:2206.12842 (2022).

diophantine-tuplesnumber-theorypell-equations

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