Question 38
Call a tuple $(b_m, b_{m+1}, \dots, b_n)$ of integers *perfect* if both following conditions are fulfilled:
1. There exists an integer $a \gt 1$ such that $b_k = a^k + 1$ for all $k = m, m+1, \dots, n$;
2. For all $k = m, m+1, \dots, n$, there exists a prime number $q$ and a non-negative integer $t$ such that $b_k = q^t$.
Prove that if $n - m$ is large enough then there is no perfect tuples, and find all perfect tuples with the maximal number of components.
1. There exists an integer $a \gt 1$ such that $b_k = a^k + 1$ for all $k = m, m+1, \dots, n$;
2. For all $k = m, m+1, \dots, n$, there exists a prime number $q$ and a non-negative integer $t$ such that $b_k = q^t$.
Prove that if $n - m$ is large enough then there is no perfect tuples, and find all perfect tuples with the maximal number of components.