Let $S$ be a set of positive integers such that no element of $S$ is a perfect square. Is it true that $\sum_{s_i \in S} \sqrt{s_i}$ is always irrational?
Motivation. Suppose the length of the circumference of a polygon whose nodes are located on lattice points is an integer. I'm trying to figure out whether this implies that the lengths of all its sides must be integers as well.
Edit: This is a slightly more general question than this one (in particular, primes versus non-squares), but appears to be answered in the same way.
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