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I saw this question in our textbook

A great physicist of the century (P.A.M. Dirac) loved playing with numerical values of Fundamental constants of nature. This led him to an interesting observation. Dirac found that from the basic constants of atomic physics ($c$, $e$, mass of electron, mass of proton) and the gravitational constant $G$. He could arrive at a number with the dimension of time. Further, it was a very large number, its magnitude being close to the present estimate on the age of the universe (~15 billion years). From the table of fundamental constants, try to see if you too can construct this number (or any other interesting number you can think of). If it's coincidence with the age of universe were significant, what would this imply for the constancy of fundamental constants?

The answer comes out to be 6 billion years, which is not close to the age of the universe yet in many places it is said that it is approximately the age of the universe.

I've also talked to people but no one has yet given me an answer that made me understand this.

Please give me an explanation about this.

Also which equation is this? I haven't been able to find it online.

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    $\begingroup$ I think the idea is the number is of the same order of magnitude (i.e. within a factor 10) of the age of the universe, and this is interesting. $\endgroup$ Sep 5, 2019 at 7:59
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    $\begingroup$ Related post: physics.stackexchange.com/q/500308/2451 $\endgroup$
    – Qmechanic
    Sep 5, 2019 at 8:03
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    $\begingroup$ Remember that the paper was in the early days of observational cosmology, so the exact age was fairly uncertain. The argument is an order of magnitude argument rather than trying to get an exact number (yet then concludes that those two big numbers truly are equal...) $\endgroup$ Sep 5, 2019 at 10:43
  • $\begingroup$ You might find the following helpful: Dirac large numbers hypothesis $\endgroup$ Sep 5, 2019 at 12:18

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From the table of fundamental constants, Dirac derived the following: \begin{aligned} &\left(\frac{1}{4 \pi \varepsilon_{0}}\right)^{2} \times \frac{e^{4}}{m_{e}^{2} m_{p} c^{3} G} \\ =& \frac{\left(9 \times 10^{9}\right)^{2} \times\left(1.6 \times 10^{-19}\right)^{4}}{\left(9.1 \times 10^{-31}\right)^{2} \times 1.67 \times 10^{-27} \times\left(3 \times 10^{8}\right)^{3} \times 6.67 \times 10^{-11}} \\ =& 2.13 \times 10^{16} \mathrm{~s} \end{aligned} which does not seem to be of the order of the age of the universe. Both the physical constants and the age of the universe have changed since the time of Dirac. Therefore Dirac hypothesis is obsolete. (https://en.m.wikipedia.org/wiki/Dirac_large_numbers_hypothesis)

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