What are these numbers trying to do?

$1^{st}$ Method : Without Logarithm

$29^{a} = 31^{b} = 37^{d} = 41^{d} = 1363783^{23}$

$29 = 1363783^{\frac{23}{a}}--------------------(*)$

$31 = 1363783^{\frac{23}{b}}--------------------(**)$

$37 = 1363783^{\frac{23}{c}}--------------------(***)$

$41 = 1363783^{\frac{23}{d}}--------------------(****)$

$(*)\cdot(**)\cdot(***)\cdot(****):$

$1363783 = 1363783^{23(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d})}$

$1 = 23(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d})$

$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = \frac{1}{23} \approx \boxed{0.04348}$

$--------------------------------$

$2^{nd}$ Method : With Logarithm

Let $1363783^{23} = x$

$a = log_{29}x; \frac{1}{a} = log_{x}29--------------------(*)$

$b = log_{31}x; \frac{1}{b} = log_{x}31--------------------(**)$

$c = log_{37}x; \frac{1}{c} = log_{x}37--------------------(***)$

$d = log_{41}x; \frac{1}{a} = log_{x}41--------------------(****)$

$(*) + (**) + (***) + (****):$

$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = log_{x}29 + log_{x}31 + log_{x}37 + log_{x}41$

$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = log_{x}1363783$

$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = \frac{1}{23} \approx \boxed{0.04348}$