A full April Fool’s story about e^(π√163) On April 1, 1975, readers of Scientific American opened Martin Gardner’s “Mathematical Games” column and found a bombshell. Gardner, with the straightest possible typewriter face, announced that, e^(π√163), was not merely close to an integer. It was an integer. Exactly. The integer was 262537412640768744. Not approximately. Not to within experimental error. Exactly. The decimal point, Gardner implied, was a stubborn illusion maintained by inferior calculators. The exact value begins: e^(π√163) = 262537412640768743.999999999999250072597198185688879353856337336990862707537410… That is not an integer. It misses by about 7.5 × 10^(−13). Which is tiny. If you owe someone e^(π√163) dollars and you pay them 262537412640768744 dollars, you have overpaid by less than a trillionth of a dollar. Most accountants would forgive you. Most computers would not even notice. In 1975, a pocket calculator asked to compute e^(π√163) would often display: 262537412640768744. The calculator was not lying. It was rounding. But to a reader in a hopeful mood, it looked like proof. Gardner’s column encouraged that hope. He wrote as if Ramanujan himself had known the exact integer, as if the near-miss had finally been explained, as if the decimal tail was just dust in the machinery. One reader claimed he had checked it on a mainframe and got the integer exactly. Another wrote that her calculator showed 262537412640768744.0, and therefore all mathematics textbooks must be revised. A graduate student reportedly spent a weekend trying to prove that e^(π√163) was rational, then algebraic, then at least not transcendental. By Sunday night he had filled three notebooks and eaten nothing but toast. Why is it so close? The answer is beautiful, and it is the reason the hoax worked. The number 163 is the largest of the Heegner numbers: 1, 2, 3, 7, 11, 19, 43, 67, 163. These are special imaginary quadratic fields with class number one. For 163, the modular j-invariant takes an enormous integer value: j((1 + √−163)/2) = −262537412640768000. Then the j-function’s q-expansion gives e^(π√163) = 262537412640768744 − 196884e^(−π√163) + 21493760e^(−2π√163) − ⋯. The first correction is already about 7.5 × 10^(−13). The next is unimaginably smaller. So the number is not an integer, but it is an integer plus a vanishing whisper. This is why Ramanujan noticed it. He was not fooled into thinking it was exactly an integer. He was fascinated by how nearly it was one. The number is often called Ramanujan’s constant, though Ramanujan never claimed it was exact. Gardner knew all this and the column had been an April Fool’s hoax. e^(π√163) is not an integer. In fact, it is not even algebraic. By the Gelfond–Schneider theorem, e^(π√163) = (−1)^(−i√163), and since −1 is algebraic and −i√163 is algebraic irrational, the number is transcendental. It cannot be an integer. It cannot be a root of any polynomial with integer coefficients. It is as far from being a rational integer as a number can be, while still dressing up like one on a calculator screen.