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Greatest Prime divisor problem

Source: Irish MO

May 10, 2012

number theory unsolvednumber theory

Problem Statement

A sequence of primes

a_n

is defined as follows:

a_1 = 2

, and, for all n \geq 2,

a_n

is the largest prime divisor of

a_1a_2...a_{n-1} + 1

. Prove that

a_n \neq 5

for all n. I'm presuming it must involve proving it's never equal to 0 mod 5, but I don't know what to do. Thanks