We generalize and solve a problem regarding the divisibility of a sequence defined as the lower integer part of powers of the largest root of a polynomial equation. Two solutions are presented using various root finding, root extraction as well as algebraic techniques. We rely on the periodic properties of linear recurrences and powers of integers modulo primes and apply methods for estimating power sums and for determining limit points of the fractional part of powers of the largest root. We also present examples and discuss numerical and computational concerns.
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