MEMO 2018 ekipno problem 7


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8. rujna 2018.
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Let a_1, a_2, \ldots be a sequence of positive integers such that a_1 = 1  \text{   and   }  a_{k+1} = a_k^3+1, \forall k \in\mathbb{N} Prove that for every prime number p of the form 3t+2, where t is a non-negative integer, there exists a positive integer n such that a_n is divisible by p.

Izvor: Srednjoeuropska matematička olimpijada 2018, ekipno natjecanje, problem 7