e^(π√163) ≈ 262537412640768743.9999999999992... The gap to the nearest integer is about 7.5 × 10⁻¹³. Twelve consecutive 9s after the decimal. The number is transcendental, but the near-miss is exact and profound.
All 9 Heegner numbers (1,2,3,7,11,19,43,67,163) produce near-integers via e^(π√d). The effect gets stronger with larger d because the j-invariant of the relevant elliptic curve is closer to an integer.
Ramanujan's constant is e raised to the power π√163. Its decimal expansion begins 262537412640768743.999999999999925… The string of twelve 9s after the decimal point is not a coincidence. It comes from deep in the theory of complex multiplication and elliptic curves.
The reason lies in a remarkable fact: 163 is a Heegner number. For such special integers, the j-invariant of a particular elliptic curve is very close to a perfect cube. This forces e^(π√163) to be extraordinarily close to an integer. It misses by less than 10⁻¹².
Martin Gardner famously reported in 1975 that e^(π√163) was exactly an integer. He did so on April 1st. The joke worked because the number is so convincingly close that many readers believed it. The true value is transcendental, but the near-miss is genuine and profound.
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.