The Heegner Number 163

The number e^(π√163) ≈ 262,537,412,640,768,743.99999999999925 is almost — but not quite — an integer. This near-miss is explained by the theory of complex multiplication: 163 is the largest Heegner number, a prime for which the imaginary quadratic field ℚ(√−163) has class number exactly 1. Ramanujan had spotted the astonishing near-integer empirically. Its full explanation required decades of deep algebraic number theory. The convergence to an integer is so extreme that a pocket calculator cannot distinguish it from one.