Reduce intermediate sizes in arithmetic-term computation of Pell coordinate sums

Determine whether the coordinate sums used to recover the fundamental solution of Pell’s equation can be obtained by an arithmetic term with substantially smaller intermediate integers than those produced by the five constructions in the paper.

Background

The paper constructs fixed straight-line programs that recover the fundamental solution of x² − d y² = 1 from coordinate sums over an initial square of lattice points. These constructions use packed integers whose bit lengths are exponential in an a priori bound for log X₁, where X₁ is the first Pell coordinate, even though the output itself has only O(√d log d) bits under Hua’s bound.

The authors explicitly leave unresolved whether the coordinate sums can be computed by an arithmetic term whose intermediate values are substantially smaller. This concerns the size efficiency of the packing and summation stage, independently of the paper’s operation-count results.