 Solution

Assume $|a{|}_2>1$, equivalently $v_2(a)<0$, and writeFor $f(Y)=Y^2-(1+4/a^4)$ at $Y=1$,The strong form of Hensel lemma therefore gives $y\in {\mathbb{Q}}_2$ with $y^2=1+4/a^4$. Taking $X=a^{2}y$ yields $X^2=a^4+4$. Hence every $a$ with $|a{|}_2>1$, and in particular every sufficiently large $a$, works.