<s>[INST] Prove that $\sqrt 2 + \sqrt 3 + \sqrt 5$ is irrational. Note that my attempted method below is distinct from the solutions in this question.
I also know this is generally true for $\sqrt p + \sqrt q + \sqrt r$ where $p,q,r$ are prime, but I am asking if my particular method works.
Prove that $\sqrt 2 + \sqrt 3 + \sqrt 5$ is irrational.
Assume that $\sqrt 2 + \sqrt 3 + \sqrt 5 = \frac pq$ for some $p,q \in \mathbb Z$, in lowest terms, with $q \neq 0$. Then at most one of $p$ and $q$ can be even.  Then
\begin{align*}
\sqrt 2 + \sqrt 3 + \sqrt 5 &= \frac pq\\
\\
(\sqrt 2 + \sqrt 3 + \sqrt 5)^2 &= \left(\frac pq \right)^2\\
\\
2(5+\sqrt 6 + \sqrt{10} + \sqrt{15})&=\frac{p^2}{q^2}\\
\\
2q^2(5+\sqrt 6 + \sqrt{10} + \sqrt{15})&=p^2
\end{align*}
which shows that $p^2$ is even, so $p$ is even and let $p=2k$ for some $k\in \mathbb Z$. Substituting this we get
\begin{align*}
2q^2(5+\sqrt 6 + \sqrt{10} + \sqrt{15})&=(2k)^2\\
\\
q^2(5+\sqrt 6 + \sqrt{10} + \sqrt{15})&=2k^2\\
\\
q^2 &= 2 \left(\frac{k^2}{5+\sqrt 6 + \sqrt{10} + \sqrt{15}}\right),
\end{align*}
which shows as well that $q$ is even, a contradiction. Hence $\sqrt 2 + \sqrt 3 + \sqrt 5$ is irrational.
 [/INST] There are infinite primes of the form $p=120k+61$ by Dirichlet's theorem. For any of them we have that $3$ and $5$ are quadratic residues, since $p\equiv 1\pmod{15}$, while $2$ is not a quadratic residue since $p\equiv 5\pmod{8}$. Assume that $\sqrt{2}+\sqrt{3}+\sqrt{5}=\frac{a}{b}$ and take a prime $p$ of the previous form with $p> \max(5,b)$. Since $\left(\frac{2}{p}\right)=-1$ while $\left(\frac{3}{p}\right)=\left(\frac{5}{p}\right)=1$, the minimal polynomial of $\sqrt{2}+\sqrt{3}+\sqrt{5}$ over $\mathbb{F}_p$ has degree $2$. In particular $\sqrt{2}+\sqrt{3}+\sqrt{5}$ cannot be an algebraic number with degree $1$ over $\mathbb{Q}$, i.e. a rational number.
Alternative approach. Always assuming $\sqrt{2}+\sqrt{3}+\sqrt{5}=q\in\mathbb{Q}$ we have
$$ 8+2\sqrt{15}=(\sqrt{3}+\sqrt{5})^2 = (q-\sqrt{2})^2 = (q^2+2)-2q\sqrt{2},$$
$$ 2\sqrt{15}+2q\sqrt{2} = (q^2-6),$$
$$ (60+8q^2) + 8q \sqrt{30} = (q^2-6)^2, $$
$$ \sqrt{30} = \frac{(q^2-6)^2-(60+8q^2)}{8q} $$
but the last line implies $\sqrt{30}\in\mathbb{Q}$ which we know not to be the case.
Both approaches can be easily extended for proving that $\sum_{k=1}^{n}\sqrt{p_k}\not\in\mathbb{Q}$ for any collection of primes $p_k$.
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