Question 58
It is said that a positive integer $n \gt 1$ has the property ($p$) if in its prime factorization
$$
n = p_1^{\alpha_1} \cdots p_j^{\alpha_j}
$$
at least one of the prime factors $p_1, \dots, p_j$ has the exponent equal to 2.
a) Find the largest number $k$ for which there exist $k$ consecutive positive integers that do not have the property ($p$).
b) Prove that there is an infinite number of positive integers $n$ such that $n, n + 1$ and $n + 2$ have the property ($p$).
$$
n = p_1^{\alpha_1} \cdots p_j^{\alpha_j}
$$
at least one of the prime factors $p_1, \dots, p_j$ has the exponent equal to 2.
a) Find the largest number $k$ for which there exist $k$ consecutive positive integers that do not have the property ($p$).
b) Prove that there is an infinite number of positive integers $n$ such that $n, n + 1$ and $n + 2$ have the property ($p$).