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Infinitely many terms divisible by 1986

Source: Canadian Mathematical Olympiad - 1986 - Problem 5.

July 3, 2011
pigeonhole principlemodular arithmeticinductionalgebra unsolvedalgebra

Problem Statement

Let u1u_1, u2u_2, u3u_3, …\dots be a sequence of integers satisfying the recurrence relation un+2=un+12−unu_{n + 2} = u_{n + 1}^2 - u_n. Suppose u1=39u_1 = 39 and u2=45u_2 = 45. Prove that 1986 divides infinitely many terms of the sequence.
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