A necessary condition for $2p$ to be congruent for a prime $p\equiv 5(\mathrm{m}\mathrm{o}\mathrm{d}8)$

In this article, we consider primes $p\equiv 5(\mathrm{m}\mathrm{o}\mathrm{d}8)$ and are able to prove that $p\equiv 5(\mathrm{m}\mathrm{o}\mathrm{d}16)$ if $2p$ is a congruent number.