# Sum of square root of non perfect square positive integers is always irrational?

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.

• – vadim123 Jul 22 '15 at 3:56
• The source material link in the above blog is broken. See p. 87 of math.harvard.edu/hcmr/issues/2.pdf – vadim123 Jul 22 '15 at 4:02
• How would the radicals cancel out otherwise, if all terms are positive? – Gary. Jul 22 '15 at 4:02
• – Chris Culter Jul 22 '15 at 4:05
• @Irvan: But in your problem statement, $S$ is a collection of positive integers? Sorry if I am missing something. – Gary. Jul 22 '15 at 4:12