A Numerical Mystery From the 19th Century Finally Gets Solved
In the early 1950s, a group of researchers at the Institute for Advanced Study embarked on a high-tech project. At the behest of John von Neumann and Herman Goldstine, the physicist Hedvig Selberg programmed the IAS’s 1,700-vacuum-tube computer to calculate curious mathematical sums whose origins stretched back to the 18th century.
The sums were related to quadratic Gauss sums, named for the famed mathematician Carl Friedrich Gauss. Gauss would choose some prime number p, then sum up numbers of the form e x (2iπn2)/p. Since their inception, quadratic Gauss sums have proved invaluable for tasks like counting solutions to certain types of equations. “It turns out that Gauss sums are magical, that they just do wonderful things for God knows what reason,” said Jeffrey Hoffstein, a mathematician at Brown University.
In the mid-19th century, the German mathematician Ernst Eduard Kummer was toying with a close relative to these. Kummer noticed that they tended to collect near particular values to a surprising degree—a keen observation that would lead to centuries of inquiry in number theory.
