Why is $n_1 \sqrt{2} +n_2 \sqrt{3} + n_3 \sqrt{5} + n_4 \sqrt{7} $ never zero? Here $n_i$ are integral numbers, and not all of them are zero.
It is natural to conjecture that similar statement holds for even more prime numbers. Namely, 
$$ n_1 \sqrt{2} +n_2 \sqrt{3} + n_3 \sqrt{5} + n_4 \sqrt{7} + n_5 \sqrt{11} +n_6 \sqrt{13} $$ is never zero too.
I am asking because this is used in some numerical algorithm in physics

If $p_1,\dots,p_n$ are prime numbers with $p_i\ne p_j$ for $i\ne j$, then the field extension $\mathbb Q\subset\mathbb Q(\sqrt{p_1},\dots,\sqrt{p_n})$ has degree $2^n$ and a basis is given by the set $$\{1,\sqrt{p_{i_1}\cdots p_{i_k}}:1\le i_1<\cdots<i_k\le n, 1\le k\le n\}.$$ See Proving that $\left(\mathbb Q[\sqrt p_1,\dots,\sqrt p_n]:\mathbb Q\right)=2^n$ for distinct primes $p_i$.
