Question 2
Let $n$ be a natural number. Permutation $a_1, a_2, \dots, a_n$ of numbers $1, 2, \dots, n$ is called square (cubic), if for each natural number $1 \le i \le n - 1$, $a_i a_{i+1} + 1$ is a perfect square (cube).
a) Prove that for infinitely many natural numbers $n$ there exists at least one square permutation of numbers $1, 2, \dots, n$.
b) Prove that for no natural number $n$ there exists a cubic permutation of numbers $1, 2, \dots, n$.
a) Prove that for infinitely many natural numbers $n$ there exists at least one square permutation of numbers $1, 2, \dots, n$.
b) Prove that for no natural number $n$ there exists a cubic permutation of numbers $1, 2, \dots, n$.