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a_{n+1} = a_n^3 + 1999, exists at most 1 perfect square

Source: Austrian - Polish 1999 APMC

May 4, 2020
recurrence relationSequencePerfect Squarenumber theory

Problem Statement

A sequence of integers (an)(a_n) satisfies an+1=an3+1999a_{n+1} = a_n^3 + 1999 for n=1,2,....n = 1,2,.... Prove that there exists at most one nn for which ana_n is a perfect square.
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