What is Ramanujan's Constant?

e^(π√163): terrifyingly close to a whole number
…744 integer e^(π√163) …743.9999999999993 gap ≈ 7.5×10⁻¹³
Table of Heegner numbers and how close e to the pi root is t
d (Heegner) e^(π√d) distance to int. 19 884736744 ~0.000022 43 884736743.9999… ~0.000002 67 147197952743.999… ~10⁻³ 163 262537…743.99999… ~7.5×10⁻¹² 163 is the largest Heegner number. Its near-integer is the most dramatic 12 nines after the decimal.
Related topics
Pi E Transcendental Numbers
Pi (π) → Gelfond's Constant → Euler's Number e → Euler's Identity → Taylor Series →
Key facts about Ramanujan

Srinivasa Ramanujan (1887-1920) was a self-taught Indian mathematician who produced extraordinary results. His 1914 series 1/pi = (2*sqrt(2)/9801) * sum of (4n)!(1103+26390n)/((n!)^4 * 396^(4n)) adds about 8 decimal digits per term and remains the basis of modern pi computation. His partition function formula was the first exact result for p(n). Ramanujan's constant e^(pi*sqrt(163)) ≈ 262537412640768743.99999999999925 is nearly an integer due to properties of the j-function.

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Want to test your knowledge?
Question
Give a simpler example of the same phenomenon with a smaller Heegner number.
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