These days I usually wander around on WolframAlpha to experiment and discover many trivial but curious calculations or mathematical relations. Recently, I have randomly discovered a strange closeness of a particular value of Gamma function and the famous Catalan's constant K (note that it is sometimes denoted as $G$ or $C$ instead)
$Γ(\sqrt{3})-K=-0.0008632972041326...≈0$
If we replace the Catalan's constant by the number $Γ(\sqrt{3})$ in the mathematical contexts, the error due to rounding will be only -0.0942% for each single replacement. This very-near-to-zero result seems to come out of the blue. It would not be incorrect to interpret it as a mathematical coincidence only. However, I think it is possible that this closeness relation is merely a consequence from very deep and sophisticated mathematical theories - much like the explanations for the Euler's lucky number or the Ramanujan constant phenomena. It is likely that a satisfied explanation to the approximation I have mentioned is currently out of reach. While many of us strongly believe that the Catalan's constant $K$ is transcendental number - I still think of the possibility that $K$ may satisfies a minimal non-trivial polynomial equation with very large rational coefficients. If this was the case, I will call that polynomial $H(x)$, so that when $x=K$ then $H(x)=0$. From the closeness relation above, we can use this polynomial to arrive at another "good" approximation: $H(Γ(\sqrt{3})≈0$
The question:
Have any mathematical context/ paper ever mentioned this mathematical coincidence? I think it deserves some studies and attention from the mathematical experts of the field (May be the Transcendental number theory? Or the Theory of Special Functions?). I have tried to find the sources for this relation everywhere - I have even looked for it at this - which is a rich collection of mathematical coincidences. But then I came back empty-handed.
Addendum
While this almost-integer result is not as impressive as the one above, the approximation is however tighter: $(\frac{K}{(Γ(\sqrt{π})+Γ(\frac{1}{\sqrt{π}}))K-2})^3=30.999955617339...≈31$ The error of deviation from being integer is only -0.000143%