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

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.

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.

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.