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The Ramanujan constant is an extremely close almost-integer, equal to \(e^{\pi\sqrt{163}} \approx 262,537,412,640,768,743.9999999999992500725971981\).[1] 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 Heegner number 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

  1. Ramanujan Constant