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 ?.
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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.
Euler was the one to prove that all even perfect numbers are of the form $$2^{n-1} ( 2^n - 1 )$$ This is known as the Euclid-Euler Theorem, and says that we just have to prove that $2^n-1$ is prime. Your number is $$2^{30} ( 2^{31} - 1 )$$ Which Euler got by just plugging integer $n$ into the equation above. Now, we have that $$2^{31}-1=2\,147\,483\,647$$ is small enough to be checked for primality by hand. Euler did this by utilizing modular properties of the Mersenne Primes, as @Chappers and others note in the comments above. In essence Euler lowered the number of cases to check considerably by noting that if $2^{31}-1$ was composite with divisor $p$ then either $p \equiv 1 \pmod {248}$ or $p \equiv 63 \pmod {248}$