MHB Irrationality of sum of roots of primes.

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The discussion explores the irrationality of sums of square roots of prime numbers, starting with established cases like $\sqrt{2}$ and $\sqrt{2} + \sqrt{3}$. It proposes that sums of the form $\sqrt{p_1} + \sqrt{p_2} + \sqrt{p_3} + \ldots + \sqrt{p_n}$ are irrational for all n, based on a method used for three primes. However, the method fails for four primes, prompting a request for alternative approaches. The conversation highlights the linear independence of square roots of primes over the rationals as a key point. Overall, the thread seeks to deepen understanding of the irrationality of sums involving prime roots.
caffeinemachine
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I observed the following:

1) $\sqrt{2}$ is irrational.

2) $\sqrt{2}+\sqrt{3}$ is irrational(since its square is irrational).

3) $\sqrt{2}+\sqrt{3}+\sqrt{5}$ is irrational(assume its rational and is equal to $r$. Write $r- \sqrt{5}=\sqrt{2} + \sqrt{3}$. Now square both the sides and its obvious from here).

So I am thinking may be $\sqrt{p_1} + \sqrt{p_2} + \sqrt{p_3} + \ldots + \sqrt{p_n}$ is irrational for all $n$, where $p_i$ is the $i-th$ prime.

The trick I used for (3) doesn't work for $n=4$. Any ideas anyone?
 
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