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Exponentially growing sequence of integers

Source: Czech and Slovak Olympiad 1978, National Round, Problem 6

October 11, 2024
number theoryalgebraSequencePerfect Square

Problem Statement

Show that the number pn=(3+52)n+(3−52)n−2p_n=\left(\frac{3+\sqrt5}{2}\right)^n+\left(\frac{3-\sqrt5}{2}\right)^n-2 is a positive integer for any positive integer n.n. Furthermore, show that the numbers p2n−1p_{2n-1} and p2n/5p_{2n}/5 are perfect squares ((for any positive integer n).n).
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