Perfect numbers, Wieferich primes and the solutions of $\left(\genfrac{}{}{0px}{}{2n}{n}\right)\equiv 2^n<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>n$

In this article we focus on the solutions of a congruence equation: $\left(\genfrac{}{}{0px}{}{2n}{n}\right)\equiv 2^{n}n$. Using the main result of this article and the SageMath software, we improve largely the number of known solutions. Furthermore, we prove that some famous numbers like even perfect numbers and Wieferich primes are connected to solutions of this equation.