Sums of square roots

Asked Jan 23 '20 at 15:32

Active Jan 23 '20 at 15:37

Viewed 48 times

Let $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_m$ be two (finite) lists of positive integers such that no $a_i$ or $b_j$ is a perfect square ($m$ may or may not be the same as $n$). Can the following equality hold?

$$\sum_{i = 1}^n \sqrt{a_i} \stackrel{?}{=} \sum_{j = 1}^m \sqrt{b_j}.$$

I just came up with this question, so I don't know what sort of techniques may be used to dis/prove this, or if there are any trivial solutions that can be excluded (other than $a_i = b_i$ up to re-indexing).

Edit: Suppose also that all $a_i$'s and $b_j$'s are required to be relatively prime.

edited Jan 23 '20 at 15:37

asked Jan 23 '20 at 15:32

taninamdar

2,662

Whoops, I forgot to include square roots. Thanks! – taninamdar Jan 23 '20 at 15:33
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What about$$\sqrt{12}=\sqrt{3}+\sqrt{3}$$ – Peter Foreman Jan 23 '20 at 15:34
Hmm. Maybe those are the trivial cases that I was thinking of. What if all $a_i$'s and $b_j$'s are required to be relatively prime? – taninamdar Jan 23 '20 at 15:35
@taninamdar Or square free. I think that's a more reasonable and interesting requirement. – Arthur Jan 23 '20 at 15:36
@Arthur could you provide an answer with that requirement? – taninamdar Jan 23 '20 at 15:38
I'm not sure. In that case it does sound to me like only trivially equal sums should be equal, but I don't really know. – Arthur Jan 23 '20 at 15:42
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Related: https://math.stackexchange.com/questions/30687/the-square-roots-of-different-primes-are-linearly-independent-over-the-field-of – Jaap Scherphuis Jan 23 '20 at 15:45

Sums of square roots

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