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sequence of largest prime factor of product of previous terms

Source: Croatian MO 2004 2nd Grade P3

April 8, 2021
number theory

Problem Statement

The sequence (pn)n∈N(p_n)_{n\in\mathbb N} is defined by p1=2p_1=2 and, for n≥2n\ge2, pnp_n is the largest prime factor of p1p2⋯pn−1+1p_1p_2\cdots p_{n-1}+1. Show that pn≠5p_n\ne5 for all nn.
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