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.
Original sources
IBM: the original challenge and solution
IBM's July 2010 page includes Henry Shin in its list of solvers.