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[a^1788] and [a^1988] divisible by 17

Source: IMO Shortlist 1988, Problem 7, IMO Longlist 10

October 22, 2005
algebrapolynomialmodular arithmeticnumber theoryDivisibilityIMO Shortlist

Problem Statement

Let a a be the greatest positive root of the equation x^3 \minus{} 3 \cdot x^2 \plus{} 1 \equal{} 0. Show that [a1788] \left[a^{1788} \right] and [a1988] \left[a^{1988} \right] are both divisible by 17. Here [x] [x] denotes the integer part of x. x.
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