Question 40
Let $S=\{a_{1}, \ldots, a_{n}\}$ be a finite set of positive integers of size $n \geq 1$, and let $T$ be the set of all positive integers that can be expressed as sums of perfect powers (including $1$) of distinct numbers in $S$, meaning
$$
T=\left\{\sum_{i=1}^{n} a_{i}^{e_{i}} \mid e_{1}, e_{2}, \ldots, e_{n} \geq 0\right\}
$$
Show that there is a positive integer $N$ (only depending on $n$) such that $T$ contains no arithmetic progression of length $N$.
$$
T=\left\{\sum_{i=1}^{n} a_{i}^{e_{i}} \mid e_{1}, e_{2}, \ldots, e_{n} \geq 0\right\}
$$
Show that there is a positive integer $N$ (only depending on $n$) such that $T$ contains no arithmetic progression of length $N$.