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Part of 2018 Philippine MO

Problems(1)

Prime linear recurrences are constant

Source: Philippines MO 2018/2

8/4/2018
Suppose a1,a2,…a_1, a_2, \ldots is a sequence of integers, and dd is some integer. For all natural numbers nn, \begin{align*}\text{(i)} |a_n| \text{ is prime;} && \text{(ii)} a_{n+2} = a_{n+1} + a_n + d. \end{align*} Show that the sequence is constant.
number theory