Question 146
a) Prove that there exist integers $a$, $b$, $c$ with $(a, b, c) \neq (0,0,0)$ and $|a|,|b|,|c|\lt 10^{6}$ for which
$$
|a+b \sqrt{2}+c \sqrt{3}|\lt 10^{-11}
$$
b) Let $a$, $b$, $c$ be integers such that $(a, b, c) \neq (0,0,0)$ and $|a|,|b|,|c|\lt 10^{6}$. Prove that
$$
|a+b \sqrt{2}+c \sqrt{3}|\gt 10^{-21}
$$
$$
|a+b \sqrt{2}+c \sqrt{3}|\lt 10^{-11}
$$
b) Let $a$, $b$, $c$ be integers such that $(a, b, c) \neq (0,0,0)$ and $|a|,|b|,|c|\lt 10^{6}$. Prove that
$$
|a+b \sqrt{2}+c \sqrt{3}|\gt 10^{-21}
$$