RefacteThe mechanism
Turn a rounding condition into finite arithmetic
The useful change of viewpoint is to collect algebraic conjugates into a trace. An integer recurrence then records the quantities relevant to rounding, and a companion matrix organises their residues. The numerical-looking question becomes a finite arithmetic calculation.
What becomes elementary: Trace identities connect the original approximation problem to an exact recurrence and modular computation.
Follow the bridge from conjugates to the recurrence before using the modular calculation. The original challenge and the supplied verification materials remain linked with the study.
For another construction, explore the mechanism calculi. Edition dates, sources and reference chapters identify the scope of each study.
This solution to the July 2010 IBM Ponder This challenge moves between algebraic numbers and finite arithmetic. Traces provide an integer recurrence, and a companion matrix organizes the modular calculation.
Background
Algebraic numbers, polynomial roots, and modular arithmetic.
Read the full exposition
The complete argument is available as a PDF.
Revision history ·
Revised November 3, 2026: the opening now derives the design requirement for an exact integer sequence with a decaying error before introducing traces and a finite modular recurrence. The mathematical argument on PDF pages 2–10 is unchanged.
Original sources
IBM: the original challenge and solution
IBM's July 2010 page includes Henry Shin in its list of solvers.