Question 98
For any integer $n$ with $n \gt 1$, let $n = p_1^{a_1} \cdots p_t^{a_t}$ be its standard factorization, write
$$
\omega(n) = t, \quad \Omega(n) = \alpha_1 + \cdots + \alpha_t.
$$
Prove or disprove the following statement: Given any positive integer $k$ and any positive real numbers $\alpha$ and $\beta$, there exists a positive integer $n$ with $n \gt 1$ such that
$$
\frac{\omega(n + k)}{\omega(n)} \gt \alpha \quad \text{and} \quad \frac{\Omega(n + k)}{\Omega(n)} \lt \beta.
$$
$$
\omega(n) = t, \quad \Omega(n) = \alpha_1 + \cdots + \alpha_t.
$$
Prove or disprove the following statement: Given any positive integer $k$ and any positive real numbers $\alpha$ and $\beta$, there exists a positive integer $n$ with $n \gt 1$ such that
$$
\frac{\omega(n + k)}{\omega(n)} \gt \alpha \quad \text{and} \quad \frac{\Omega(n + k)}{\Omega(n)} \lt \beta.
$$