The smallest odd perfect number must exceed $10^{300}$. I am studying about perfect numbers from last two week and have experienced so much adventure in studying such an interesting topic. The basic sources have been Wikipedia and the book Euler: Master of us all.
After proving so many results and reading so much theory I m stuck on one of the results mentioned at the end of the the book "Euler: Master of us all".
The result is as follow:

The smallest odd perfect number must exceed $10^{300}$.

Since the name of mathematician who gave the result is not given in the book so I can't even find it on internet. I shall be highly thankful if you can give me a hint to approach for this result or can supply a direct proof. Forgive me if this result is trivial and I m missing very common thing.
Thanks.
 A: From http://mathworld.wolfram.com/OddPerfectNumber.html 

To this day, it is not known if any odd perfect numbers exist,
  although numbers up to $10^{300}$ have been checked without success,
  making the existence of odd perfect numbers appear unlikely (Brent et
  al. 1991; Guy 1994, p. 44). The following table summarizes the
  development of ever-higher bounds for the smallest possible odd
  perfect number. There is a project underway at
  http://www.oddperfect.org/ seeking to extend the limit beyond
  $10^{300}$. 
  
  
*
  
*author bound  
  
*Kanold (1957)  $10^{20}$
  
*Tuckerman (1973)   $10^{36}$ 
  
*Hagis    (1973)    $10^{50}$
  
*Brent and Cohen (1989) $10^{160}$
  
*Brent et al.   (1991) $10^{300}$
  
  
  Brent, R. P. and Cohen, G. L. "A New Bound for Odd Perfect Numbers."
  Math. Comput. 53, 431-437 and S7-S24, 1989.
Brent, R. P.; Cohen, G. L.; te Riele, H. J. J. "Improved Techniques
  for Lower Bounds for Odd Perfect Numbers." Math. Comput. 57, 857-868,
  1991.
Guy, R. K. "Perfect Numbers." §B1 in Unsolved Problems in Number
  Theory, 2nd ed. New York: Springer-Verlag, pp. 44-45, 1994.

A: The Wikipedia article gives a stronger lower bound, $10^{1500}$, which is shown in a paper by Ochem and Rao (2012) that says they obtained the improvement by modifying the method by which Brent, Cohen, and te Riele (1991) got the bound you ask about.  See the PDF here or the Math. Comp. journal page here.
A: That's hardly a "common thing".
The paper establishing the $10^{300}$ bound dates back to $1991$ and can be downloaded from the author's page: Improved techniques for lower bounds for odd perfect numbers .

Abstract
If $N$ is an odd perfect number, and $q^k$ is the highest power of $q$ dividing $N$, where $q$ is prime and $k$ is even, then it is almost immediate that $N \gt q^{2k}$. We prove here that, subject to certain conditions verifiable in polynomial time, in fact $N > q^{5k/2}$. Using this and related results, we are able to extend the computations in an earlier paper to show that $N > 10^{300}$.

See also the OddPerfect.org preaanouncement.
