How Euler showed that number :$2305843008139952128$ is perfect without using computer calculation? This number : $2305843008139952128$ is perfect as shown here and it was proved that is a perfect number by Euler without using computer , then 


My question here is:
    How Euler showed that number :$2305843008139952128$ is perfect without using computer calculation ?. 



Since the number given is $2^{30}(2^{31}-1)$, all that needs to be done is show that $2^{31}{-}1$ is prime. Euler knew that factors of this number must be of the form $k(2\cdot31)+1$, and also must be $8n\pm1$. This gives $84$ primes to check for division up to $\sqrt(2^{31}-1)$
See Modular restrictions on Mersenne divisors from the Prime Pages.
