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 | \(g_{\omega^{\omega4+4}4+\omega^{\omega4+3}2}(5)\) | \(g_{\omega^{14}9}(15)\) |