A note on the irrationality of ${\zeta }_2(5)$

In a spirit of Apéry’s proof of the irrationality of $\zeta (3)$, we construct a sequence $p_n\mathrm{/}{q}_n$ of rational approximations to the $2$-adic zeta value ${\zeta }_2(5)$ which satisfy $0<\mathrm{\mid }{\zeta }_2(5)-p_n\mathrm{/}{q}_n{\mathrm{\mid }}_2<\max \{\mathrm{\mid }{p}_n\mathrm{\mid },\mathrm{\mid }{q}_n\mathrm{\mid }{\}}^{-1-\delta }$ for an explicit constant $\delta >0$. This leads to a new proof of the irrationality of ${\zeta }_2(5)$, the result established recently by Calegari, Dimitrov and Tang using a different method. Furthermore, our approximations allow us to obtain an upper bound for the irrationality measure of this $2$-adic quantity; namely, we show that $\mu ({\zeta }_2(5))\le (16\log 2)\mathrm{/}(8\log 2-5)=20.342\dots$.