Ramanujan constant

The Ramanujan constant is an extremely close almost-integer, equal to \(e^{\pi\sqrt{163}} \approx 262,537,412,640,768,743.9999999999992500725971981\).Ramanujan Constant It came from an April fool's prank by Martin Gardner, where he claimed that \(e^{\pi\sqrt{163}}\) was actually an integer and that Ramanujan himself hypothesized this. Ramanujan had no actual involvement with the number.
The closeness of the number to an integer is not at all a coincidence; \(-163\) is a and \(e^\pi\) has important properties on the complex plane.
The integer closest to the constant, 262,537,412,640,768,744, is equal to 640,3203 + 744. Its prime factorization is 23 × 3 × 10,939,058,860,032,031.
==Approximations==
Notation 	Lower bound 	Upper bound
Scientific notation 	\(2.625\times10^{17}\) 	\(2.626\times10^{17}\)
Arrow notation 	\(86\uparrow9\) 	\(22\uparrow13\)
Steinhaus-Moser Notation 	14[3] 	15[3]
Copy notation 	2[18] 	3[18]
Taro's multivariable Ackermann function	A(3,54) 	A(3,55)
Pound-Star Notation 	#*(4,4,2)*6 	#*(1,2)*11
BEAF 	{86,9} 	{22,13}
Hyper-E notation 	2E17 	3E17
Bashicu matrix system 	(0)(0)[22635] 	(0)(0)[22636]
Hyperfactorial array notation 	19! 	20!
Fast-growing hierarchy 	\(f_2(52)\) 	\(f_2(53)\)
Hardy hierarchy 	\(H_{\omega^2}(52)\) 	\(H_{\omega^2}(53)\)
Slow-growing hierarchy 	\(g_{\omega^{\omega4+4}4+\omega^{\omega4+3}2}(5)\)	\(g_{\omega^{14}9}(15)\)
== Sources ==
Category:Numbers
Category:Class 2
Category:Eponymous numbers
Category:Humor
Category:Numbers with 7 to 29 digits
Category:Transcendental numbers
Category:Pi-related numbers
Category:E-related numbers