Question 107
$f: \mathbb{N} \to \mathbb{N}$ is an increasing function and $n$ is a natural number. Suppose that there exist prime numbers $p_1, p_2, \ldots, p_n$ and natural numbers $s_1, s_2, \ldots, s_n$ such that for every $1 \le i \le n$ the set $\{f(p_i r + s_i) \mid r = 1, 2, \ldots \}$ is an infinite arithmetic progression. Prove that there exists natural number $a$ such that
$$
f(a + 1), f(a + 2), \ldots, f(a + n)
$$
is an arithmetic progression.
$$
f(a + 1), f(a + 2), \ldots, f(a + n)
$$
is an arithmetic progression.