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q(x) to be the product of all primes less than p(x)

Source: IMO Shortlist 1995, S3

August 10, 2008
inductionalgebrapolynomialIterationSequenceIMO ShortlistHi

Problem Statement

For an integer x≥1x \geq 1, let p(x)p(x) be the least prime that does not divide xx, and define q(x)q(x) to be the product of all primes less than p(x)p(x). In particular, p(1)=2.p(1) = 2. For xx having p(x)=2p(x) = 2, define q(x)=1q(x) = 1. Consider the sequence x0,x1,x2,…x_0, x_1, x_2, \ldots defined by x0=1x_0 = 1 and xn+1=xnp(xn)q(xn) x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} for n≥0n \geq 0. Find all nn such that xn=1995x_n = 1995.
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