abstract algebra - 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