Theoretical work on such issues does exist; for example, [Dilcher 1999]

discourses on the difficulty of creating longer squaring ladders of the indicated

kind. Recently, D. Symes has discovered a (k = 4) identity, with coefficients

(a1, a2, a3, a4) as implied in the construct

$\displaystyle (((x^2-67405)^2-3525798096)^2-533470702551552000)^2-4692082091913216002$

which, as the reader may wish to verify via symbolic processing, is indeed

the product of 16 monomials! P. Carmody recently reports that many such

4-squarings cases are easy to generate via, say, a GP/Pari script.